PA/2.T -IL
/&77
[Prom the Transactions of the Connecticut Academy of Arts and Sciences,
Vol. Ill, Part 2.]
COT THE EQUILIBEIUM
OF
HETEROGENEOUS SUBSTANCES.
Second Part.
By J. WILL ARD GIBBS,
PROFESSOR OF MATHEMATICAL PHYSICS IN YALE COLLEGE, NEW HAVEN, CONN.
The Dibner Library
of the History of
Science and Technology
SMITHSONIAN INSTITUTION LIBRARIES
O 3 ' J fcrcr-
-7
On the Equilibrium of Heterogeneous Substances.
By J. WlLLARD GlBBS.
( Continued from page 248).
THE CONDITIONS OP INTERNAL AND EXTERNAL EQUILIBRIUM FOR
SOLIDS IN CONTACT WITH FLUIDS WITH REGARD TO ALL POSSIBLE
STATES OF STRAIN OF THE SOLIDS.
In treating of the physical properties of a solid, it is necessary to
consider its state of strain. A body is said to be strained when the
relative position of its parts is altered, and by its state of strain is
meant its state in respect to the relative position of its parts. We
have hitherto considered the equilibrium of solids only in the case in
which their state of strain is determined by pressures having the
same values in all directions about any point. Let us now consider
the subject without this limitation.
If x 1 , y', ?J are the' rectangular co-ordinates of a point of a solid
body in any completely determined state of strain, which we shall
call the state of reference, and x, y, z, the rectangular co-ordinates of
the same point of the body in the state in which its properties are the
subject of discussion, we may regard x, y, z as functions ofx', y', z',
the form of the functions determining the second state of strain.
For brevity, we may sometimes distinguish the variable state, to
which x, y, z relate, and the constant state (state of reference), to
which x', y\ z' relate, as the strained and the unstrained states ; but
it must be remembered that these terms have reference merely to the
change of form or strain determined by the functions which express
the relations of x, y, z and x', y', z', and do not imply any particular
physical properties in either of the two states, nor prevent their
possible coincidence. The axes to which the co-ordinates x, y, z, and
x', y', z' relate will be distinguished as the axes of X, Y, Z, and
X\ Y\ Z '. It is not necessary, nor always convenient, to regard
these systems of axes as identical, but they should be similar, i. e.,
capable of superposition.
The state of strain of any element of the body is determined by the
values of the differential coefficients of x, y, and z with respect to
x', y', and z'\ for changes in the values of x, y, z, when the differential
coefficients remain the same, only cause motions of translation of the
Trans. Qonn. Acad., Vol. III. 44 May, 1877.
344 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
body. When the differential coefficients of the first order do not
vary sensibly except for distances greater than the radius of sensible
molecular action, we may regard them as completely determining the
state of strain of any element, There are nine of these differential
coefficients, viz.,
dx dx dx ]
aW dy" ~dz n I
d JL d v d v [ ,»**
dx" ay> w y (3o4)
dz dz dz
dx 1 ' dy n dz 1 ' ,
It will be observed that these quantities determine the orientation of
the element as well as its strain, and both these particulars must be
given in order to determine the nine differential coefficients. There-
fore, since the orientation is capable of three independent variations,
which do not affect the strain, the strain of the element, considered
without regard to directions in space, must be capable of six indepen-
dent variations.
The physical state of any given element of a solid in any unvary-
ing state of strain is capable of one variation, which is produced by
addition or subtraction of heat. If we write e vi and // v/ for the
energy and entropy of the element divided by its volume in the
state of reference, we shall have for any constant state of strain
Ss v , = t drj Yl .
But if the strain varies, we may consider s vi as a function of rf y , and
the nine quantities in (354), and may write
Ss Y , = t 6> v , + X XI d~ + X YI d— t + X z , 8—
dx' dy dz'
where JT X ,, . . . Z ZI denote the differential coefficients of £ Y , taken
with respect to -^- n . . . — r The physical signification of these
quantities will be apparent, if we apply the formula to an element
which in the state of reference is a right parallelopiped having the
edges dx', dy', dz', and suppose that in the strained state the face in
which x' has the smaller constant value remains fixed, while the
opposite face is moved parallel to the axis of X. If we also suppose
DG
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 345
no beat to be imparted to the element, we shall have, on multiplying
by dx' dy' dz' ,
de Y , dx' dy' dz' = X x , 6-=-, dx' dy' dz'.
Now the first member of this equation evidently represents the work
done upon the element by the surrounding elements ; the second
member must therefore have the same value. Since we must regard
the forces acting on opposite faces of the elementary parallelopiped as
equal and opposite, the whole work done will be zero except for the
(%/QC
face which moves parallel to X. And since 6-^—, dx' represents the
distance moved by this face, X XI dy' dz' must be equal to the com-
ponent parallel to X of the force acting upon this face. In general,
therefore, if by the positive side of a surface for which x' is constant
we understand the side on which x has the greater value, we may say
that Xx, denotes the component parallel to X oi the force exerted by
the matter on the positive side of a surface for which x' is constant
upon the matter on the negative side of that surface per unit of the
surface measured in the state of reference. The same may be said,
mutatis mutandis, of the other symbols of the same type.
It will be convenient to use 2 and 2' to denote summation with
respect to quantities relating to the axes X, Y~, Z, and to the axes
X', Y' , Z' , respectively. With this understanding we may write
dX, = t di h , + 2 2' (X x , d~y (356)
This is the complete value of the variation of s v , for a given element
of the solid. If we multiply by dx' dy' dz', and take the integral for
the whole body, we shall obtain the value of the variation of the total
energy of the body, when this is supposed invariable in substance.
But if we suppose the body to be increased or diminished in substance
at its surface (the increment being continuous in nature and state
with the part of the body to which it is joined), to obtain the com-
plete value of the variation of the energy of the body, we must add
the integral
J'£ v ,6X'Ds'
in which Ds' denotes an element of the surface measured in the state
of reference, and dN' the change in position of this surface (due to
the substance added or taken away) measured normally and out-
ward in the state of reference. The complete value of the variation
of the intrinsic energy of the solid is therefore
346 J. W. G-ibbs — Equilibrium of Heterogeneous Substances.
ffftSr/ Yl dx'dy'dz' + fff ^t{x xi 6~]\ dx'dy'dz 1 + fe r , SN'Ds'. (357)
This is entirely independent of any supposition in regard to the
homogeneity of the solid.
To obtain the conditions of equilibrium for solid and fluid masses
in contact, we should make the variation of the energy of the whole
equal to or greater than zero. But since we have already examined
the conditions of equilibrium for fluids, we need here only seek the
conditions of equilibrium for the interior of a solid mass and for the
surfaces where it comes in contact with fluids. For this it will be
necessary to consider the variations of the energy of the fluids only
so far as they are immediately connected with the changes in the
solid. We may suppose the solid with so much of the fluid as is in
close proximity to it to be enclosed in a fixed envelop, which is
impermeable to matter and to heat, and to which the solid is firmly
attached wherever they meet. We. may also suppose that in the
narrow space or spaces between the solid and the envelop, which are
filled with fluid, there is no motion of matter or transmission of heat
across any surfaces which can be generated by moving normals to the
surface of the solid, since the terms in the condition of equilibrium
relating to such processes may be cancelled on account of the internal
equilibrium of the fluids. It will be observed that this method is
perfectly applicable to the case in which a fluid mass is entirely
enclosed in a solid. A detached portion of the envelop will then be
necessary to separate the great mass of the fluid from the small
portion adjacent to the solid, which alone we have to consider. Now
the variation of the energy of the fluid mass will be, by equation
(13),
fH SDrf -f F p dDv + ^ 1 f F M> 8Dm ±i (358)
where f F denotes an integration extending over all the elements of
the fluid (within the envelop), and 2 t denotes a summation with
regard to those independently variable components of the fluid of
which the solid is composed. Where the solid does not consist of
substances which are components, actual or possible (seepage 117),
of the fluid, this term is of course to be cancelled.
If we wish to take account of gravity, we may suppose that it acts
in the negative direction of the axis of Z. It is evident that the
variation of the energy due to gravity for the whole mass considered
is simply
fffgT'Szdx'dg'dz', (359)
where g denotes the force of gravity, and F' the density of the
J. W. Gfibbs — Equilibrium of Heterogeneous Substances. 347
element in the state of reference, and the triple integration, as before,
extends throughout the solid.
We have, then, for the general condition of equilibrium,
/ dx\
ffft 6i h , dx' dy' d*' +ff/2 2' (X x , 6—j dx' dp' dz 1
4- fff.g V dz dx' dy 1 dz' -\-f £ VI SJV' Ds'
+f ¥ t SBi] -fp SDv + 2 , /> 1 SDm , ^ 0. (360)
The equations of condition to which these variations are subject, are:
(1) that which expresses the constancy of the total entropy,
fffdrjy, dx'dy' dz' +»< W ^ +f* SI >V = ° 5 ( 3 ^)
(2) that which expresses how the value of SDv for any element of
the fluid is determined by changes in the solid,
dDv =z—(<xdx + fiSy + y dz) Ds - v vi 6N 1 Ds', (362)
where a, fi, y denote the direction cosines of the normal to the
surface of the body in the state to which x, y, z relate, Ds the element
of the surface in this state corresponding to Ds' in the state of
reference, and v Y , the volume of an element of the solid divided by
its volume in the state of reference ;
(3) those which express how the values of 6Dm 1 , 6Dm 2 , etc. for
any element of the fluid are determined by the changes in the solid,
&Dm 1 = - r x ' 8N' Ds', "]
SDm 2 = - iy SJV' Ds', } (363)
etc., J
where I\, r % ', etc. denote the separate densities of the several com-
ponents in the solid in the state of reference.
Now, since the variations of entropy are independent of all the
other variations, the condition of equilibrium (360), considered with
regard to the equation of condition (361), evidently requires that
throughout the whole system
t= const. (364)
We may therefore use (361) to eliminate the first and fifth integrals
from (360). If we multiply (362) by p, and take the integrals for
the whole surface of the solid and for the fluid in contact with it, we
obtain the equation
f F p dDv = - fp (a 6x-\~ (3 dy + y dz) Ds - fp v yi W Ds', (365)
by means of which we may eliminate the sixth integral from (360).
If we add equations (363) multiplied respectively by /x 1 , /u 2 , etc.,
and take the integrals, we obtain the equation
348 J. W. Gibbs — Equilibrium, of Heterogeneous Substances.
2 1 fjz 1 SIhn 1 = -/^(/1,/y) dN'Ds', (366)
by means of which we may eliminate the last integral from (360).
The condition of equilibrium is thus reduced to the form
ff/2 2' (x x , d ~)j dx' dy' dz> +fffg V dz dx' dy' dz'
+ f£ Y , dX' Ds' -ft r, N , dX' J)s' +fp{adx + /i dy -f y dz) Ds
+ fpv Y ,dX'Ds> -f2 x 0-iJ\') dX'Ds'^0, (367)
in which the variations are independent of the equations of condition,
and in which the only quantities relating to the fluids are p and ju ly
yu 2 , etc.
Now by the ordinary method of the calculus of variations, if we
write a', /5', y' for the direction-cosines of the normal to the surface
of the solid in the state of reference, we have
fffX xl d^dx'dy'dz>
dX
—fa! X x , dx Ds' - fff—2. dx dx' dy' dz', (368)
with similar expressions for the other parts into which the first
integral in (36 7) may be divided. The condition of equilibrium is
thus reduced to the form
-fffZ 2' (~? dx) dx' dy' dz' +fffg V dz dx' dy' dz'
+/2 2' {a' X XI dx) Ds' +fp2(a dx) Ds
+/[£ V ,- ttfy.+pvy, - 2, {li.r^dX'Ds'^ 0. (369)
It must be observed that if the solid mass is not continuous
throughout in nature and state, the surface-integral in (368), and
therefore the first surface-integral in (369), must be taken to apply
not only to the external surface of the solid, but also to every surface
of discontinuity within it, and that with reference to each of the
two masses separated by the surface. To satisfy the condition of
equilibrium, as thus understood, it is necessary and sufficient that
throughout the solid mass
2 2' (^ dx) -gr>dz = 0- (370)
that throughout the surfaces where the solid meets the fluid
Ds' 2 2' (a' X x , dx) + Dsp 2 (a dx) = 0, (371)
and
^^i^+^^-^f^r/pj'.IO; (372)
and that throughout the internal surfaces of discontinuity
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 349
2 2' (a'X x , dx) t + 2 2' {a' X x , Sx) 2 = 0, (373)
where the suffixed numerals distinguish the expressions relating to
the masses on opposite sides of a surface of discontinuity.
Equation (370) expresses the mechanical conditions of internal
equilibrium for a continuous solid under the influence of gravity. If
we expand the first term, and set the coefficients of dx, dy, and dz
separately equal to zero, we obtain
dX x , dX Y , dX z ,
dx' dy' dz'
dx' dy 1 dz'
dZ x , dZ YI dZ Zl
= gF'.
dx' ^ dy' ^ dz'
The first member of any one of these equations multiplied by dx' dy'
dz' evidently represents the sum of the components parallel to one of
the axes X, Y, Z of the forces exerted on the six faces of the element
dx' dy' dz' by the neighboring elements.
As the state which we have called the state of reference is arbitrary,
it may be convenient for some purposes to make it coincide with the
state to which x, y, z relate, and the axes X', Y' , Z' with the axes
X, Y, Z. The values of X x „ . . . Z z , on this particular supposition
may be represented by the symbols X x , . . . Z z . Since
de Y , de v ,
Xy ' = ~^ and - X ' = ^3'
dy' dx'
and since, when the states x, y, z and x', y', z' coincide, and the axes
X, Y, Z, and X', Y', Z', d-^— f and d~ represent displacements
which differ only by a rotation, we must have
X Y = Y X , (375)
and for similar reasons,
Y Z = Z Y , Z X = X Z . (376)
The six quantities X x , Y Y , Z z , X Y or 3^, Y z or Z Y , and Z x orX z are
called the rectangular components of stress, the three first being the
longitudinal stresses and the three last the shearing stresses. The
mechanical conditions of internal equilibrium for a solid under the
influence of gravity may therefore be expressed by the equations
350 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
dX x , dX Y , dX 7
■— o,
dx dy dz
dY x , dT Y dY 7
dZ x , dZ Y , dZ z
dx dy dz
where F denotes the density of the element to which the other sym-
bols relate. Equations (375), (376) are rather to be regarded as
expressing necessary relations (when X x , . . . Z z are regarded as
internal forces determined by the state of strain of the solid) than as
expressing conditions of equilibrium. They will hold true of a solid
which is not in equilibrium, — of one, for example, through which
vibrations are propagated, — which is not the case with equations (377).
Equation (373) expresses the mechanical conditions of equilibrium
for a surface of discontinuity within the solid. If we set the coeffi-
cients of dx, dy, dz, separately equal to zero we obtain
(«'jr x/ -f/3'X Y ,+;/'X z J 1 -f(^X x/ +/5'X Y/ + r 'JT z ,) 2 =:0, ]
(a' Y x ,+/3' Y Y ,+y> Y ZI ) , + («' F x ,+/?' Y Y ,+ y' Y z ,) 2 =0, } (378)
[at Z Xl +/3> Z YI +y' Z z ,) t -{-(a' Z x ,+/3' Z Y ,+y' Z Z ) % =Q. J
Now when the a', fi', y' represent the direction-cosines of the normal
in the state of reference on the positive side of any surface within the
solid, an expression of the form
a' X x , + /J' X YI + / X ZI (379)
represents the component parallel to X of the force exerted upon
the surface in the strained state by the matter on the positive
side per unit of area measured in the state of reference. This is
evident from the consideration that in estimating the force upon
any surface we may substitute .for the given surface a broken one
consisting of elements for each of which either x' or y or z' is
constant. Applied to a surface bounding a solid, or any portion of a
solid which may not be continuous with the rest, when the normal is
drawn outward as usual, the same expression taken negatively repre-
sents the component parallel to X of the force exerted upon the
surface (per unit of surface measured in the state of reference) by the
interior of the solid, or of the portion considered. Equations (378)
therefore express the condition that the force exerted upon the
surface of discontinuity by the matter on one side and determined by
its state of strain shall be equal and opposite to that exerted by the
matter on the other side. Since
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 351
(a'), = - («%, {fi) t = - (/r> 2 , m, = - (/)*,
we may also write
a'(X x ,) ^'(X*,) ^/(X,,) , -a'(X x ,) s +/i'(X Y/ ) 3 +;/(X z ,) 2 ,| (38o)
etc., j
where the signs of «', /3', ;/' may be determined by the normal on
either side of the surface of discontinuity.
Equation (371) expresses the mechanical condition of equilibrium
for a surface where the solid meets a fluid. It involves the separate
equations
a X x , + /?' X Y , + / X z/ = - ap —-, ,
a! Y x , + /J' Y YI + / F z , = - /3p ~-, , (381)
the fraction -=— , denoting the ratio of the areas of the same element
of the surface in the strained and unstrained states of the solid.
These equations evidently express that the force exerted by the
interior of the solid upon an element of its surface, and determined
by the strain of the solid, must be normal to the surface and equal
(but acting in the opposite direction) to the pressure exerted by the
fluid upon the same element of surface.
If we wish to replace a and Ds by a , fi\ y\ and the quantities
which express the strain of the element, we may make use of the
following considerations. The product a Ds is the projection of the
Ds
element Ds on the Y-Z plane. Now since the ratio -=— , is indepen-
dent of the form of the element, we may suppose that it has any
convenient form. Let it be bounded by the three surfaces x' = const. ,
y' r= const., z' = const., and let the parts of each of these surfaces
included by the two others with the surface of the body be denoted
by i, M, and i\^ or by _Z7, M' and X', according as we have reference
to the strained or unstrained state of the body. The areas of Z', M',
and X' are evidently a' Ds', /3' Ds', and y' Ds' ; and the sum of the
projections of L, M and Xupon any plane is equal to the projection
of Ds upon that plane, since L, M, and X with Ds include a solid
figure. (In propositions of this kind the sides of surfaces must be
distinguished. If the normal to Ds falls outward from the small
solid figure, the normals to L, M, and X must fall inward, and vice
Trans. Conn. Acad., Vol. III. 45 May, 1877.
352 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
versa). Now U is a right-angled triangle of which the perpendicular
sides may be called dy' and dz'. The projection of L on the Y-Z
plane will be a triangle, the angular points of which are determined
by the co-ordinates
. dy , dz , dy ' . dz _ ,
y, >; v + w %, * + w <Jy; y+^ d *> H--^*;
the area of such a triangle is
' dy dz dz dy\ , ,
~dy~' W ~ ^dy 1 W) V '
or, since \ dy' dz' represents the area of 77,
dy dz dz dy \
=-,-£-)« Ds'.
K dy' dz' dy 1 dz' ,
(That this expression has the proper sign will appear if we suppose
for the moment that the strain vanishes.) The areas of the pro-
jections of M 2l\\& iv"upon the same plane will be obtained by chang-
ing y', z' and a' in this expression into z', x', and /?', and into x', y',
and y' . The sum of the three expressions may be substituted for
aDs in (381).
We shall hereafter use 2' to denote the sum of the three terms
obtained by rotary substitutions of quantities relating to the axes
X' , Y', Z', (i. e., by changing x\ y\ z' into y', z', x', and into z', x\ y',
with similar changes in regard to a', ft', y', and other quantities
relating to these axes,) and 2 to denote the sum of the three terms
obtained by similar rotary changes of quantities relating to the axes
X, Y, Z. This is only an extension of our previous use of these
symbols.
With this understanding, equations (381) may be reduced to the
form
y x0 ^ 1 I \dy' dz' dy' dz' J j ' y (382)
etc. J
The formula (372) expresses the additional condition of equilibrium
which relates to the dissolving of the solid, or its growth without
discontinuity. If the solid consists entirely of substances which are
actual components of the fluid, and there are no passive resistances
which impede the formation or dissolving of the solid, 6N' may have
either positive or negative values, and we must have
e Y , - tr?v,+.pvv,= 2 1 (ji/ i r,'). (383)
But if some of the components of the solid are only possible com-
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 353
ponents (see page 117) of the fluid, 6R' is incapable of positive
values, as the quantity of the solid cannot be increased, and it is
sufficient for equilibrium that
^-i^+^^^^r/). (384)
To express condition (383) in a form independent of the state of
reference, we may use € v , Vv, r t , etc., to denote the densities of
energy, of entropy, and of the several component substances in the
variable state of the solid. We shall obtain, on dividing the equa-
tion by v v „
s v -t? ?v +p=2 1 ( Mi r i ). (385)
It will be remembered that the summation relates to the several
components of the solid. If the solid is of uniform composition
throughout, or if we only care to consider the contact of the solid
and the fluid at a single point, we may treat the solid as composed of
a single substance. If we use /x 1 to denote the potential for this
substance in the fluid, and F to denote the density of the solid in the
variable state, {T\ as before denoting its density in the state of
reference,) we shall have
«vi - t?7v,+pv v ,z= Mi r \ ( 38 6)
and
e Y — tr/ v -\-p=i /x 1 F. (38V)
To fix our ideas in discussing this condition, let us apply it to the
case of a solid body which is homogeneous in nature and in state of
strain. If we denote by €, ?/, v, and m, its energy, entropy, volume,
and mass, we have
s — t f] -f- p v = u j m. (388)
Now the mechanical conditions of equilibrium for the surface where
a solid meets a fluid require that' the traction upon the surface deter-
mined by the state of strain of the solid shall be normal to the sur-
face. This condition is always satisfied with respect to three surfaces
at right angles to one another. In proving this well known proposi-
tion, we shall lose nothing in generality, if we make the state of
reference, which is arbitrary, coincident with the state under discus-
sion, the axes to which these states are referred being also coincident.
We shall then have, for the normal component of the traction per unit
of surface across any surface for which the direction-cosines of the
normal are a, /J, y, [compare (379), and for the notation _Z" X , etc.,
page 349,]
354 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
S=a{aX x + pX Y +yX z )
+ P(aY x + pY Y +yY z )
+ y (aZ x + p Z Y + y Z z ),
or, by (375), (376),
-\-2apjL Y + 2PyY z + 2'ya Z x . (389)
We may also choose any convenient directions for the co-ordinate
axes. Let us suppose that the direction of the axis of JT is so chosen
that the value of S for the surface perpendicular to this axis is as
great as for any other surface, and that the direction of the axis of Y
(supposed at right angles to X) is such that the value of S for the
surface perpendicular to it is as great as for any other surface
passing through the axis of X Then, if we write ~ , — , — for
da dp ' dy
the differential coefficients derived from the last equation by treating
a, p, and y as independent variables,
da "" T afj »r ~r dy vy - y
when
a da -f- P dp -\- y dy = 0,
and
a=l, P = 0, y=0.
That is,
dS . dS
— - = 0, and -=- = 0,
dp dy
when
a=l, P=0, y = 0.
Hence,
X Y = 0, and Z x = 0.
Moreover,
dS T/) , dS -,
when
a^O, da= 0,
Pdp+y dy = 0,
and
P=l, y = 0.
Hence
Y z =o.
(390)
(391)
Therefore, when the co-ordinate axes have the supposed directions,
which are called the principal axes of stress, the rectangular com-
ponents of the traction across any surface (a, P, y) are by (379)
aX x , P Y Y , yZ z . (392)
Hence, the traction across any surface will be normal to that
surface, —
(1), when the surface is perpendicular to a principal axis of stress ;
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 355
(2), if two of ike principal tractions X x , F Y , Z z are equal, when the
surface is perpendicular to the plane containing the two correspond-
ing axes, (in this case the traction across any such surface is equal to
the common value of the two principal tractions) ;
(3), if the principal tractions are all equal, the traction is normal and
constant for all surfaces.
It will he observed that in the second and third cases the position of
the principal axes of stress are partially or wholly indeterminate, (so
that these cases may be regarded as included in the first,) but the
values of the principal tractions are always determinate, although not .
always different.
If, therefore, a solid which is homogeneous in nature and in state of
strain is bounded by six surfaces perpendicular to the principal axes ,
of straiiCthe mechanical conditions of equilibrium for these surfaces
may be satisfied by the contact of fluids having the proper pressures,
[see (381),] which will in general be different for the different pairs of
opposite sides, and may be denoted by p', p", p'". (These pressures
are equal to the principal tractions of the solid taken negatively.)
It will then be necessary for equilibrium with respect to the tendency
of the solid to dissolve that the potential for the substance of the
solid in the fluids shall have values jjl x \ /i/', fj. t '" determined by the
equations
s -trf+p'v = Pi'm, (393)
8 — tr}+p"v — }A- x " m, (394)
8— tr/+p'"v = fx 1 '"m. (395)
These values, it will be observed, are entirely determined by the
nature and state of the solid, and their differences are equal to the
differences of the corresponding pressures divided by the density of
the solid.
It may be interesting to compare one of these potentials, as ///,
with the potential (for the same substance) in a fluid of the same
temperature t and pressure p' which would be in equilibrium with the
same solid subjected on all sides to the uniform pressure p'. If we
write [e\,, [r/] P , , \v\,, and [Mi] P > for the values which £, 77, v, and
yUj would receive on this supposition, we shall have
0L< - * [*< +P' K" = Uhl> m> (396)
Subtracting this from (393), we obtain
€ - [e] pl — t? ? -\-t \n\„ +y v - p' [v] pl = fi x m — [m^, m. (397)
Now it follows immediately from the definitions of energy and entropy
356 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
that the first four terms of this equation represent the work spent
upon the solid in bringing it from the state of hydrostatic stress to the
other state without change of temperature, and p' v — p' \v] p , evi-
dently denotes the work done in displacing a fluid of pressure p
surrounding the solid during the operation. Therefore, the first
number of the equation represents the total work done in bringing
the solid when surrounded by a fluid of pressure p' from the state of
hydrostatic stress p' to the state of stress p', p" , p"'. This quantity is
necessarily positive, except of course in the limiting case when
p' = p" — p'". If the quantity of matter of the solid body be unity,
the increase of the potential in the fluid on the side of the solid on
which the pressure remains constant, which will be necessary to
maintain equilibrium, is equal to the work done as above described.
Hence, /// is greater than [/^J^, , and for similar reasons, pt /' Is
greater than the value of the potential which would be necessary for
equilibrium if the solid were subjected to the uniform pressure^", and
/</" greater than that which would be necessary for equilibrium if
the solid were subjected to the uniform pressure p"'. That is, (if we
adapt our language to what we may regard as the most general case,
viz., that in which the fluids contain the substance of the solid but
are not wholly composed of that substance,) the fluids in equilibrium
with the solid are all supersaturated with respect to the substance
of the solid, except when the solid is in a state of hydrostatic stress; so
that if there were present in any one of these fluids any small frag-
ment of the same kind of solid subject to the hydrostatic pressure of
the fluid, such a fragment would tend to increase. Even when no
such fragment is present, although there must be perfect equilibrium
so far as concerns the tendency of the solid to dissolve or to increase
by the accretion of similarly strained matter, yet the presence of the
solid which is subject to the distorting stresses, will doubtless
facilitate the commencement of the formation of a solid of hydrostatic
stress upon its surface, to the same extent, perhaps, in the case of
an amorphous body, as if it were itself subject only to hydrostatic
stress. This may sometimes, or perhaps generally, make it. a necessary
condition of equilibrium in cases of contact between a fluid and an
amorphous solid which can be formed out of it that the solid at the
surface where it meets the fluid shall be sensibly in a state of hydro-
static stress.
But in the case of a crystairline solid, subjected to distorting stresses *■
and in contact with solutions satisfying the conditions deduced above,
although crystals of hydrostatic stress would doubtless commence to
J. ~W. Gibbs — Equilibrium of Heterogeneous Substances. 357
form upon its surface (if the distorting stresses and consequent
supersaturation of the fluid should he carried too far), before
they would commence to he formed within the fluid or on
the surface of most other bodies, yet within certain limits the
relations expressed by equations (393)-(395) must admit of realiza-
tion, especially when the solutions are such as can be easily super-
saturated.*
It may be interesting to compare the variations of p, the pressure
in the fluid which determines in part the stresses and the state of
strain of the solid, with other variations of the stresses or strains in
the solid, with respect to the relation expressed by equation (388).
To examine this point with complete generality, we may proceed in
the following manner.
Let us consider so much of the solid as has in the state of reference
the form of a cube, the edges of which are equal to unity, and
parallel to the co-ordinate axes. We may suppose this body to be
homogeneous m nature and in state of strain both in its state of
reference and in its variable state. (This involves no loss of
generality, since we may make the unit of length as small as we
choose.) Let the fluid meet the solid on one or both of the surfaces
for which Z' is constant. We may suppose these surfaces to remain
perpendicular to the axis of Z in the variable state of the solid, and the
edges in which y' and z' are both constant to remain parallel to the
axis of X. It will be observed that these suppositions only fix the
position of the strained body relatively to the co-ordinate axes, and
do not in any way limit its state of strain.
It follows from the suppositions which we have made that
dz dz dy
W = ,== ' ^7 = const = °' ^- con8t - = °; ( 398 )
and
^~ n xr . „ dx dy
T z ,= 0, T zl =z0, Z zl = —p-~ T -f- l .
ax ay
(399)
(400)
Hence, by (355),
7 ^ 7 i t^ 7 <% x i x- 7 dx ■ T7- 7 dy dx dy , dz
dEy—tdTjy^X^d—-, -\-A Y ,d-— -4- T Yl d-~, ~p-r-, -~ d -—,
dx 1 dy' dy' l dx' dy' dz'
Again, by (388),
* The effect of distorting stresses in a solid on the phenomena of crystallization and
liquefaction, as well as the effect of change of hydrostatic pressure common to the
solid and liquid, was first described by Professor James Thomson. See Trans. R. S.
Eclin., vol. xvi, p. 515 ; and Proc. Roy. Soc, vol. xi, p. 473, or Phil. Mag., S. 4, vol.
xxiv, p. 395.
358 J. W. Gibhs — Equilibrium of Heterogeneous Substances.
ds=zt drj -j- ?/ dt — p dv — v dp + w dfx x . (401)
Now the suppositions which have been made require that
dx dv. dz
• = S-3?W (402)
and
, dy dz dx dz dx dy dx dy dz
~ dy' dz' dx' dz' dx' dy' dx' dy' dz '
Combining equations (400), (401), and (403), and observing that
€ YI and 7/y, are equivalent to £ and ?/, we obtain
i? dt — v dp -j- nt djA x
( __. dy dz\ -,dx , ^ , dx , /^ , dz dx \ ^dy fl K
={ A ^W S )**? + Al "**7 +( r *'+* *" *?>^ < 404 >
The reader will observe that when the solid is subjected on all sides
to the uniform normal pressure p, the coefficients of the differentials
in the second member of this equation will vanish. For the expres-
sion -—-, -— represents the projection on the Y-Z plane of a side of
Chi] Ct£
the parallelopiped for which x' is constant, and multiplied by p it
will be equal to the component parallel to the axis of X of the total
pressure across this side, i. e.,it will be equal to JT X , taken negatively.
The case is similar with respect to the coefficient of d~^- f ; and JT Y ,
evidently denotes a force tangential to the surface on which it acts.
It will also be observed, that if we regard the forces acting upon the
sides of the solid parallelopiped as composed of the hydrostatic pres-
sure p together with addition forces, the work done in any infinitesimal
variation of the state of strain of the solid by these additional forces
will be represented by the second member of the equation.
We will first consider the case in which the fluid is identical in
substance with the solid. We have then, by equation (97), for a
mass of the fluid equal to that of the solid,
t/ f dt ■*- v F dp + m dfx 1 = 0, (405)
r/ F and v F denoting the entropy and volume of the fluid. By subtrac-
tion we obtain
- to -rj)dt + to — v ) d P
/ , dy dz\ 7 dx . _ -.dx / dz dx\ dy
{Jit 1 ff'Kl Cm'} l
Now if the quantities j-„ -=-„ —-, remain constant, we shall have
for the relation between the variations of temperature and pressure
which is necessary for the preservation of equilibrium
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 359
* = *ZJ? = *!!LZi!, (407)
dp Vv—V Q
where Q denotes the heat which would be absorbed if the solid body
should pass into the fluid state without change of temperature or
pressure. This equation is similar to (181), which applies to bodies
dt
subject to hydrostatic pressure. But the value of -=- will not gener-
ally be the same as if the solid were subject on all sides to the uni-
form normal pressure p; for the quantities v and rf (and therefore
Q) will in general have different values. But when the pressures on
all sides are normal and equal, the value of j will be the same,
whether we consider the pressure when varied as still normal and
f/nt* ff'K* ciu
equal on all sides, or consider the quantities -=-„ -=-„ -=—, as constant.
But if we wish to know how the temperature is affected if the pres-
sure between the solid and fluid remains constant, but the strain of
the solid is varied in any way consistent with this supposition, the
differential coefficients of t with respect to the quantities which ex-
press the strain are indicated by equation (406). These differential
coefficients all vanish, when the pressures on all sides are normal and
,.„. . , «. . dt . dx dx dy
equal, but the differential coefficient -5-, when -~ n — -„ -^- f are con-
stant, or when the pressures on all sides are normal and equal, van-
ishes only when the density of the fluid is equal to that of the solid.
The case is nearly the same when the fluid is not identical in sub-
stance with the solid, if we suppose the composition of the fluid to
remain unchanged. We have necessarily with respect to the fluid
a K> = [irl m dt + Wh m dp * (+08)
where the index (f) is used to indicate that the expression to which
it is affixed relates to the fluid. But by equation (92)
(pT =-(£-)" , and mr =(*f . (409)
\dt/p,m \dm 1 /t,p,m \dp/t,m \dm l /t,p,m
Substituting these values in the preceding equation, transposing
terms, and multiplying by m, we obtain
* A suffixed m stands here, as elsewhere in this paper, for all the symbols m,, m 2 .
etc., except such as may occur in the differential coefficient.
Trans. Conn. Acad., Vol. III. 46 May, 1817.
360 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
( drj \ (F) 7 / dv V F) T , x
i ( 3— dfc - »w ( -= — ) db + m du. = 0. (410)
\dm 1 /t, p,m \dm 1 /t,p,rn
ml
n I — - ) and
\dm 1 /t,p, m
By subtracting this equation from (404) we may obtain an equation
similar to (406), except that in place of r/ F and v F we shall have the
expressions
/ dv y F >
\dmjt, p, m'
The discussion of equation (406) will therefore apply mutatis mutan-
dis to this case.
We may also wish to find the variations in the composition of the
fluid which will be necessary for equilibrium when the pressure p or
. . dx dx dy . _. .
the quantities -=-„ -=— „ -=—, are varied, the temperature remaining
constant. If we know the value for the fluid of the quantity repre-
sented by t, on page 142 in terms of t, p, and the quantities of the
several components m ls m 2 , m 3 , etc., the first of which relates to the
substance of which the solid is formed, we can easily find the value
of pt x in terms of the same variables. Now in considering variations
in the composition of the sol-id, it will be sufficient if we make all but
one of the components variable. We may therefore give to m 1 a
constant value, and making t also constant, we shall have
Mu-iY F) 7 , Pdii.\<n 7 , /^.f
dfJL A z=[-P) dp + l -— ' dm 2 + ( -— * ) dm s + etc.
71 \dp /t, m \dm 2 /t, p, m \dm 3 /t, p, m
Substituting this value in equation (404), and cancelling the term
containing dt, we obtain
\ m (pr _A dp+m (py> dni2
( \ dp It, m ) \dm 2 /t, p, m
/duA m _ , /V dy dz\ 7 dx
+ m WJ tl „ m dm * + eta = ( x - + " w a?) '*?
. ._ 7 dx /„ dz dx\ dy
This equation shows the variation in the quantity of any one of the
components of the fluid (other than the substance which forms the
.,.,-11-1 • • d r dx dx dy .
solid) which will balance a variation of p, or of — ■„ -=— „ -=- „ with re-
spect to the tendency of the solid .to dissolve.
J. W. Gribbs — Equilibrium of -Heterogeneous Substances. 361
Fundamental Equations for Solids.
The principles developed in the preceding pages show that the
solution of problems relating to the equilibrium of a solid, or at least
their reduction to purely analytical processes, may be made to de-
pend upon our knowledge of the composition and density of the solid
at every point in some particular state, which we have called the
state of reference, and of the relation existing between the quantities
(fit* fll* ff^
which have been represented by e Vf , y yi , — ■,, — , , . . . -p , x', y\
and z'. When the solid is in contact with fluids, a certain knowledge
of the properties of the fluids is also requisite, but only such as is
necessary for the solution of problems relating to the equilibrium of
fluids among themselves.
If in any state of which a solid is capable, it is homogeneous in its
nature and in its state of strain, we may choose this state as the state
of reference, and the relation between s Yr , ?j yt , — , , . . — , will be
independent of x', y', z'. ' But it is not always possible, even in the
case of bodies which are homogeneous in nature, to bring all the
elements simultaneously into the same state of strain. It would not
be possible, for example, in the case of a Prince Rupert's drop.
If, however, we know the relation between e Yl , Tj yi , -y-, , . . . — n
for any kind of homogeneous solid, with respect to any given state of
reference, we may derive from it a similar relation with respect to
any other state as a state of reference. For if x', y\ z' denote the
co-ordinates of points of the solid in the first state of reference, and
x", y", z" the co-ordinates of the same points in the second state of
reference, we shall have necessarily
dx dx dx" dx dy" dx dz" . .
n = 3? 3? + w a? + a? dx" et0 - (mne e 1 uatI0ns >' < 412 >
and if we write B for the volume of an element in the state (x\ y\ z")
divided by its volume in the state (x\ y', z'), we shall have
R =
dx"
dx"
dx"
dx'
dy'
dz'
ay
ay
dy"
dx'
dy'
dz'
dz"
dz"
dz"
dx'
dy'
dz'
(413)
362 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
s v , — R £ v „, 7/ v , = It Tj Y „. (414)
If, then, we have ascertained by experiment the value of € YI in terms
fjjrp ft?
of 7/v/, -y- n . . . -5-7, and the quantities which express the composi-
CLQ& OCX
tion of the body, by the substitution of the values given in (412)-
(414), we shall obtain £ v „ in terms of r/ v „, -^, . . . -^, -^, . . . -_,
and the quantities which express the composition of the body.
We may apply this to the elements of a body which may be varia-
ble from point to point in composition and state of strain in a given
state of reference (as", y", z"), and if the body is fully described in
that state of reference, both in respect to its composition and to the
displacement which it would be necessary to give to a homogeneous
solid of the same composition, for which e Vl is known in terms of r/ Vl ,
cIqc> clx
— , 1 • • • -T-, , and the quantities which express its composition, to
(MX/ Cvt£>
bring it from the state of reference («:', y\ z') into a similar and
similarly situated state of strain with that of the element of the non-
dx" dz"
homogeneous body, we may evidently regard -=-y, . . , -^-, as known
for each element of the body, that is, as known in terms of x", y", z" .
We shall then have £ ytl in terms of 7/ v „, -j—„ , . . . -yi, , x'\y",z" ; and
since the composition of the body is known in terms of as", y", z", and
the density, if not given directly, can be determined from the density
of the homogeneous body in its state of reference (x', y', z'), this is
sufficient for determining the equilibrium of any given state of the
non-homogeneous solid.
An equation, therefore, which expresses for any kind of solid, and
with reference to any determined state of reference, the relation
dx dz
between the quantities denoted by £ VI , r/ Yl , -=- } , . . . -y-, , involving
also the quantities which express the composition of the body, when
that is capable of continuous variation, or any other equation from
which the same relations may be deduced, may be called a funda-
mental equation for that kind of solid. It will be observed that the
sense in which this term is here used, is entirely analogous to that in
which we have already applied the term to fluids and solids which
are subject only to hydrostatic pressure.
cloc (1%
When the fundamental equation between £ v# , ;/ v ,, -=-„ . . . -=-, is
ijjx> dj%
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 363
known, we may obtain by differentiation the values of t, JT X ,, . . . Z Zi
in terms of the former quantities, which will give eleven independent
relations between the twenty-one quantities
£v '' ^ v/ ' cfe" ' " " dz' ' ' ' " * v 415 )
which are all that exist, since ten of these quantities are independent.
All these equations may also involve variables which express the
composition of the body, when that is capable of continuous varia-
tion.
If we use the symbol tp y , to denote the value of tp (as defined on
pages 144, 145) for any element of a solid divided by the volume of
the element in the state of reference, we shall have
tf>y,= e v , — tr} Vl . (416)
The equation (356) may be reduced to the form
6Yv, = - rfr, St + 2 2' (X x , d~)j. (41V)
Therefore, if we know the value of ip Y , in terms of the variables t,
-=—, ,...-=-,, together with those which express the composition of
the body, we may obtain by differentiation the values of 7/ v „ JT X/ ,
. . . Z z , in terms of the same variables. This will make eleven inde-
pendent relations between the same quantities as before, except that
we shall have i/j y , instead of £ v , . Or if we eliminate ip x ., by means
of equation (416), we shall obtain eleven independent equations be-
tween the quantities in (415) and those which express the composi-
tion of the body. An equation, therefore, which determines the
value of ip YI , as a function of the quantities t, -=—, ,...-=-,, and the
quantities which express the composition of the body when it is capa-
ble of continuous variation, is a fundamental equation for the kind of
solid to which it relates.
In the discussion of the conditions of equilibrium of a solid, we
might have started with the principle that it is necessary and sufficient
for equilibrium that the temperature shall be uniform throughout the
whole mass in question, and that the variation of the force-function
(-ip) of the same mass shall be null or negative for any variation in
the state of the mass not affecting its temperature. We might have
assumed that the value of if: for any same element of the solid is a
364 J. W. Gfibbs — Equilibrium of Heterogeneous Substances.
function of the temperature and the state of strain, so that for con-
stant temperature we might write
^„ = ^(x x , <?§,),
the quantities X Xl , . . . Z z , being defined by this equation. This
would be only a formal change in the definition of X x , , . . . Z Zl and
would not affect their values, for this equation holds true of X x , ,
. . . Z z , as defined by equation (355). With such data, by transfor-
mations similar to those which we have employed, we might obtain
similar results.* It is evident that the only difference in the equa-
tions would be that //v, would take the place of € vt , and that the
terms relating to entropy would be wanting. Such a method is
evidently preferable with respect to the directness with which the
results are obtained. The method of this paper shows more distinctly
the rdle of energy and entropy in the theory of equilibrium, and can
be extended more naturally to those dynamical problems in which
motions take place under the condition of constancy of entropy of
the elements of a solid (as when vibrations are propagated through a
solid), just as the other method can be more naturally extended to
dynamical problems in which the temperature is constant. (See
note on page 145.)
We have already had occasion to remark that the state of strain
of any element considered without reference to directions in space is
capable of only six independent variations. Hence, it must be possi-
ble to express the state of strain of an element by six functions of
da*, dx
-=—, , . . . -j- f , which are independent of the position of the element.
For these quantities we may choose the squares of the ratios of
elongation of lines parallel to the three co-ordinate axes in the state
of reference, and the products of the ratios of elongation for each
pair of these lines multiplied by the cosine of the angle which they
include in the variable state of the solid. If we denote these quanti-
ties by A, B, C, a, b, c, we shall have
* For an example of this method, see Thomson and Tait's Natural Philosophy, vol. i,
p. 705. With regard to the general theory of elastic solids, compare also Thomson's
Memoir "On the Thermo-elastic and Thermo-magnetic Properties of Matter" in the
Quarterly Journal of Mathematics, vol. i, p. 51 (1855), and Green's memoirs on the
propagation, reflection, and refraction of light in the Transactions of the Cambridge
Philosophical Society, vol. vii.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 365
„ (dx dx\ , _.„ (dx dx \ „ /cfo efcc \ ,
«= 3 (aj?a?> »= 2 ta5?> C=2 W> (419)
The determination of the fundamental equation for a solid is thus
reduced to the determination of the relation between £ Vl , rj y , , A, JB,
C, a, b, c, or of the relation between ip YI , t, A, B, C, or, b, c.
In the case of isotropic solids, the state of strain of an element, so
far as it can affect the relation of £ yi and 7/ v , , or of tp v , and t, is capa-
ble of only three independent variations. This appears most dis-
tinctly as a consequence of the proposition that for any given strain
of an element there are three lines in the element which are at right
angles to one another both in its unstrained and in its strained
state. If the unstrained element is isotropic, the ratios of elonga-
tion for these three lines must w T ith 7/ v , determine the value £ v , , or
with t determine the value of ip v , .
To demonstrate the existence of such lines, which are called the
principal axes of strain, and to find the relations of the elongations
cine ftT
of such lines to the quantities — , , . . . — -, , we may proceed as fol-
lows. The ratio of elongation r of any line of which a', fi\ y' are
the direction-cosines in the state of reference is evidently given by
the equation
(dx , dx dx A 2
, (dz . , dz „, , dz A 2
+{*t t/ +pp+wn i (42o)
Now the proposition to be established is evidently equivalent to this
— that it is always possible to give such directions to the two sys-
tems of rectangular axes X\ Y', Z, and JT, Y, Z, that
dx dx dy ~\
— o — — - —
ay'— ' dz'~ ' dz'~ ' I
dy dz dz '
dx'~ > dx~' — °> dy'—°' J
We may choose a line in the element for which the value of r is at
least as great as for any other, and make the axes of JTand X' par-
allel to this line in the strained and unstrained states respectively.
366 J. W. Gibbs — Equilibrium of Heterogeneous /Substances.
Theu t? = ' s = °- ( 422 )
d(r 2 ) d(r 2 ) d(r 2 )
Moreover, if we write ~~f, -W, -W for the differential coeffi-
da dp dy
cients obtained from (420) by treating a', /3', y' as independent
variables,
d(r 2 ) _ , , d(r z ) 7 _, , c/(r 2 ) _ ,
when a' <?a' + ff d/3' -f 7' dy' = 0,
and a' = 1, /3' = 0, 7' = 0.
That is, -V-/ = 0, and -V-/ = 0,
when a' = 1, /5' = 0, 7' zz: 0.
Hence, ^ = 0, ^=0. (423)
Therefore a line of the element which in the unstrained state is per-
pendicular to X' is perpendicular to X in the strained state. Of all
such lines we may choose one for which the value of r is at least as
great as for any other, and make the axes of Y' and Y parallel to
this line in the unstrained and in the strained state respectively.
Then
dz
¥ , = o; («4)
and it may easily be shown by reasoning similar to that which has
just been employed that
| = 0- (42.,)
Lines parallel to the axes of X', Y\ and Z in the unstrained body
will therefore be parallel to X, Y, and Z in the strained body, and
the ratios of elongation for such lines will be
dx dy dz
dx' ' dy' ' dz' '
These lines have the common property of a stationary value of
the ratio of elongation for varying directions of the line. This
appears from the form to which the general value of r 2 is reduced by
the positions of the co-ordinate axes, viz.,
-©- + ©> + ©>•
J. W. Gibbs — Equilibrium of Heterogeneous /Substances. 367
Having thus proved the existence of lines, with reference to any
particular strain, which have the properties mentioned, let us pro-
ceed to find the relations between the ratios of elongation for these
lines (the principal axes of strain) and the quantities -=-, , ... -^-,
under the most general supposition with respect to the position of
the co-ordinate axes.
For any principal axis of strain we have
d(r 2 ) _ , , d(r 2 ) ... , d{r 2 ) _ ,
when
a' da' + f3' d/3' + y' dy' = 0,
the differential coefficients in the first of these equations being deter-
mined from (420) as before. Therefore,
(426)
(427)
2^_)_ 1 d(r*) _ 1 d(r 2 )
a' da 1 ' ~ p' ~d/3 T ~ y' ~dy'~'
From (420) we obtain directly
a'djr 2 ) fi'djr 2 ) y' d{r 2 ) _ g
2 da'' + 2 d/3' + 2 #/ ~~ r '
From the two last equations, in virtue of the necessary relation
a' 2 -j- (3' 2 -f r '2 = l, we obtain
? da' ~ ar ' * dp' - P r ' F ~^/' - r r ' ( '
or, if we substitute the values of the differential coefficients taken
from (420),
, /dx\ 2 ^ /dx dx\ . /dx dx\ , ~\
" * (a?) + f 2 (as? a?) + y 3 (a? s) = a r '
, ^ /dx dx\ , Jt ^,/dx\ 2 . (dx dx\
, v (dx dx X [dx dx\
W^'r*.
If we eliminate a', fi', y' from these equations, we may write the
result in the form,
(dx dx "
'&V- H&%
dx'J
'dx dx\
dy' dx' )
y, /dx dx
\Wdx~'.
(dx\ 2
)
/dx dx
\dx d?J
'dx dx
Tkans. Conn. Acad., Yol. III.
dy' dz'
47
= 0. (430)
May, 1877.
368 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
We may write
Then
Also*
- r 6 + JEr* - Fr 2 + G = 0. (431)
— y y \ i^ x V y i^ x V ^ x ^ x y l^ x ^ x \ i -
( W/ W/ ~dx'dy~' \d^'dy>)\-
-^\(dx\ 2 (dy\ 2 (dx\ 2 (dz \ 2 _dx dx dy dy dx dx dz dz 1
l\c?a?7 \%7 \^7 \e?y'/ dx' dy' dx' dy'~~dx' dy'dx' dy'l
— 2' 2\ /^\ 3 /^y\ 2 I /dy\ 2 /dx\ 2 _ ? dx dx dy dy )
~ \\dx') \dy') ^~\dx'') \dy~') ~ M dy~' dx~' dy~' )
= 2' 2 (— ^- — ^L ^.V ( 433)
\dx' dy' dx' dy'J ' ^ '
This may also be written
F= 2' 2
\AjWJ \AJtAJ
dx' dy'
dy dy
dx' dy'
(434)
In the reduction of the value of G, it will be convenient to use the
symbol 2 to denote the sum of the six terms formed by changing
x, y, z, into y, z, x; z, x, y; x, z, y\ y, x, z; and z, y,x; and the
symbol 2 in the same sense except that the last three terms are to
3-3
be taken negatively ; also to use 2' in a similar sense with respect
3-3
to x', y,' z' ; and to use x', y', z' as equivalent to x\ y', z', except that
they are not to be affected by the sign of summation. With this
understanding we may write
^ ^, S ^ (dx dx \ „ (dx dx \ „ (dx dx\ )
e = 3 ?; 1 2 (sif(rrtf(*s)f- < 435 >
* The values of F and G given in equations (434) and (438), which are here
deduced at length, may be derived from inspection of equation (430) by means of the
usual theorems relating to the multiplication of determinants. See Salmon's Lessons
Introductory to the Modern Higher Algebra, 2d Ed., Lesson III; or Baltzer's Theorie und
Anwendung der Determinanten, § 5.
J. W. G-ibbs — Equilibrium of Heterogeneous Substances. 369
In expanding the product of the three sums, we may cancel on
account of the sign 2' the terms which do not contain all the three
3-3
expressions dx, dy, and dz. Hence we may write
, /dx dx dy dy dz dz \
— 3 _ 3 3+3 \dx.'dx' dy' dy' dz' dz'/
= 2
3 + 3
= 2
3-3
dx dy dz , /dx dy dz\ \
dx~' dy~' dz 1 3-3 W W dz') )
'dx dy dz\ , (dx dy^ dz\
dx' dy' dz 1 ) 3 - 3 \dx' dy' dz')'
Or, if we set
H=
we shall have
dx dx dx
dx' dy' dz'
dy_ dy_
dx dy'
dz dz
dx' dy'
Q=.m.
dy
dz 1
dz
w
(436)
(437)
(438)
It will be observed that F represents the sum of the squares of the
nine minors which can be formed from the determinant in (43*7), and
that E represents the sum of the squares of the nine constituents of
the same determinant.
Now we know by the theory of equations that equation (431) will
be satisfied in general by three different values of r 2 , which we may
denote by r 2 , r 2 2 , r 3 2 , and which must represent the squares of
the ratios of elongation for the three principal axes of strain ; also that
E, E, G, are symmetrical functions of r 1 2 , r 2 2 , r 3 2 , viz.,
E= r 2 +r 2 * +r 3 2 , F= r^r^ + r 2 2 ?
G = r 1 2 r 2 2 r s 2 .
+ ^3
Vi *'\ (439)
Hence, although it is possible to solve equation (431) by the use of
trigonometrical functions, it will be more simple to regard e v , as a
function of y y , and the quantities E, E, G (or If), which we have
expressed in terms of -^-, , . . . -p . Since 8 vt is a single-valued func-
tion of 77 V , and r, 2 , r 2 3 , r 3 3 (with respect to all the changes of which
the body is capable), and a symmetrical function with respect to r 2 ,
r 2 2 , r 2 , and since r x 2 , r 2 2 , r 3 3 are collectively determined without
ambiguity by the values of E, F, and If, the quantity £ v , must be a
3Y0 J.W. Gibbs — Equilibrium of Heterogeneous Substances.
single-valued function of t/ v/ , E, E, and H. The determination of
the fundamental equation for isotropic bodies is therefore reduced to
the determination of this function, or (as appears from similar con-
siderations) the determination of >p Vl as a function of t, E, E, and H.
It appears from equations (489) that E represents the sum of the
squares of the ratios of elongation for the principal axes of strain,
that i? represents the sum of the squares of the ratios of enlargement
for the three surfaces determined by these axes, and that G repre- .
sents the square of the ratio of enlargement of volume. Again, equa-
tion (432) shows that E represents the sum of the squares of the
ratios of elongation for lines parallel to X\ Y', and Z' ; equation
(434) shows that E represents the sum of the squares of the ratios of
enlargement for surfaces parallel to the planes X'- Y', Y'-Z', Z'-X' ;
and equation (438), like (439), shows that G represents the square
of the ratio of enlargement of volume. Since the position of the
co-ordinate axes is arbitrary, it follows that the sum of the squares of
the ratios of elongation or enlargement of three lines or surfaces
which in the unstrained state are at right angles to one another, is
otherwise independent of the direction of the lines or surfaces.
Hence, ^E and }E are the mean squares of the ratios of linear elon-
gation and of superficial enlargement, for all possible directions in
the unstrained solid.
There is not only a practical advantage in regarding the strain as
determined by E, E, and H, instead of E, E, and G, because H is
more simply expressed in terms of —,,... ~ , but there is also a
certain theoretical advantage on the side of E, E, IT. If the sys-
tems of co-ordinate axes X, Y, Z, and X', Y', Z', are either iden-
tical or such as are capable of superposition, which it will always be
convenient to suppose, the determinant H will always have a posi-
tive value for any strain of which a body can be capable. But it is
possible to give to x, y, z such values as functions of x\ y\ z' that H
shall have a negative value. For example, we may make
x = x', y — y\ z — - z'. (440)
This will give JZ= — 1, while
x — x\ y = y', z = z' (441)
will give Hz= 1. Both (440) and (441) give (? = ]. ISTow although
such a change in the position of the particles of a body as is repre-
sented by (440) cannot take place while the body remains solid, yet
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 371
a method of representing strains may be considered incomplete,
which confuses the cases represented by (440) and (441).
We may avoid all such confusion by using E, E, and H to repre-
sent a strain. Let us consider an element of the body strained which in
the state (a?', y\ z') is a cube with its edges parallel to the axes of
JC', Y', Z', and call the edges dx', dy' , dz according to the axes to
which they are parallel, and consider the ends of the edges as posi-
tive for whicli the values of x ! , y', or z' are the greater. Whatever
may be the nature of the parallelopiped in the state (x, y, z) which
corresponds to the cube dx\ dy', dz' and is determined by the quanti-
ties ~y— f , ... J-, , it may always be brought by continuous changes
to the form of a cube and to a position in which the edges dx', dy'
shall be parallel to the axes of X and Y, the positive ends of the
edges toward the positive directions of the axes, and this may be done
without giving the volume of the parallelopiped the value zero,
and therefore without changing the sign of H. Now two cases are
possible; — the positive end of the edge dz' may be turned toward the
positive or toward the negative direction of the axis of Z. In the
first case, H is evidently positive ; in the second, negative. The
determinant ijTwill therefore be positive or negative, — we may say,
if we choose, that the volume will be positive or negative, — according
as the element can or cannot be brought from the state (x, y, z) to the
state (x', y', z') by continuous changes without giving its volume the
value zero.
If we now recur to the consideration of the principal axes of strain
and the principal ratios of elongation r 15 r 2 , r 3 , and denote by U 1 ,
U 2 , TI Z and £7",', U~ 2 ', U z ' the principal axes of strain in the strained
and unstrained element respectively, it is evident that the sign of r- ti
for example, depends upon the direction in U 1 which we regard as
corresponding to a given direction in XT t '. If we choose to associate
directions in these axes so that r 15 r 2 , r 3 shall all be positive, the
positive or negative value of ^Twill determine whether the system of
axes V 1 , U 2 , U 3 is or is not capable of superposition upon the sys-
tem J/"/, U 2 \ U z ' so that corresponding directions in the axes shall
coincide. Or, if we prefer to associate directions in the two systems
of axes, so that they shall be capable of superposition, corresponding
directions coinciding, the positive or negative value of If will deter-
mine whether an even or an odd number of the quantities r t , r 2 , r 3
are negative, In this case we may write
372 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
r, r, r„ = H=
dx
dx
dx
dx'
dy>
dz'
dy_
dx'
dy
dj'
dy
dz'
dz
dx'
dz
dy'
dz
dz
(442)
It will be observed that to change the signs of two of the quantities
r i> r -zi r 3 i s simply to give a certain rotation to the body without
changing its state of strain.
Whichever supposition we make with respect to the axes J7 lf U 2 ,
£/" 3 , it is evident that the state of strain is completely determined by
the values E, F, and H, not only when we limit ourselves to the
consideration of such strains as are consistent with the idea of solidity,
but also when we regard any values of — , ,...—, as possible.
(JjQu W&
Approximative Formulae. — For many purposes the value of s v , for
an isotropic solid may be represented with sufficient accuracy by the
formula
s v , = i' + e' E+f F-\- h' 11, (443)
where i', e',f, and h' denote functions of rf v ,j or the value of ip Y , by
the formula
fa, = i + e E+/F+ h H, (444)
where i, e,f\ and h denote functions of t. Let us first consider the
second of these formulae. Since E, F, and H are symmetrical func-
tions of r,, r 2 , r 3 , if fa, is any function of t, E, F, IT, we must have
dfa, dfa, dfa,
dr x dr 2 dr 3 '
d 2 fa,_d*fa,_d 2 fa,
dr s 2 >
y
(445)
dr x 2 dr 2
d 2 fa, _ d*fa, _ d 2 fa,
dr t dr 2 dr 2 dr 3 ~ dr 3 dr x J
whenever r l =r 2 =r 3 . Now i, <?,/, and h may be determined (as
functions of t) so as to give to
dfa, d 2 fa, d 2 fa,
^ v " dr^' dr~J~> dr~ x dV 2
their proper values at every temperature for some isotropic state of
strain, which may be determined by any desired condition. We
shall suppose that they are determined so as to give the proper
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 373
values to if,\,, etc., when the stresses in the solid vanish. If we
denote by r the common value of r,, r 2 , r 3 which will make the
stresses vanish at any given temperature, and imagine the true value
of ip v , , and also the value given by equation (444) to be expressed in
terms of the ascending powers of
7"i -?V ** 2 — r o> r a~ r o, ( 446 )
it is evident that the expressions will coincide as far as the terms of
the second degree inclusive. That is, the errors of the values of tp v ,
given by equation (444) are of the same order of magnitude as the
cubes of the above differences. The errors of the values of
dtpy, dip YI dipy,
dr x ' dr 2 ' dr 3
will be of the same order of magnitude as the squares of the same
differences. Therefore, since
difjy, dipy, dr x dipy, dr 2 dip v , dr 3 ua*\
dx dr x dx dr 2 dx dr 3 dx ^ k '
dx' dx' dx' dx'
whether we regard the true value of ip y , or the value given by equa-
tion (444), and since the error in (444) does not affect the values of
dr t dr 2 dr 3
dx ' dx ' dx '
dx' dx' dx'
which we may regard as determined by equations (431), (432), (434),
(437) and (438), the errors in the values of _Zx, derived from (444)
will be of the same order of magnitude as the squares of the differ-
ences in (446). The same will be true with respect to JE-y, , J£ z , , Y x ,
etc., etc.
It will be interesting to see how the quantities e, /, and h are
related to those which most simply represent the elastic properties of
isotropic solids. If we denote by V and II the elasticity of volume
and the rigidity* (both determined under the condition of constant
temperature and for states of vanishing stress), we shall have as
definitions :
v= ~ v \£)t> when v=:r o 3v, > ( 44s )
* See Thomson and Tait's Natural Philosophy, vol.' i, p. 111.
374 J. W. Gibhs— Equilibrium of Heterogeneous Substances.
where p denotes a uniform pressure to which the solid is subjected,
v its volume, and v' its volume in the state of reference ; and
C dy' \ %7 J
when ^ = %_^_ r f (449)
dx' dy'~ dz' — r "
dx_ __dx_dy _dy _dz _ dz _ j
<%' dfe' cfe' cfe' efe' dy' ' J
Now when the solid is subject to uniform pressure on all sides, if
we consider so much of it as has the volume unity in the state of
reference, we shall have
r x — r % =r 3 = v^, (450)
and by (444) and (439),
ip YI =i + 3ev w -{-3fv^-\-hv. (451)
Hence, by equation (88), since ip YI is equivalent to ip,
and by (448),
P = (&)t = 2ev ~ i + 4 f vi + h > ( 452 )
igr £^ F:= ~ f ^ + i/>0 ' (454)
To obtain the value of JR, in accordance with the definition (449),
we may suppose the values of E, E, and H given by equations (432),
(434), and (437) to be substituted in equation (444). This will give
for the value of H
r o H=2e + 4fr K (455)
Moreover, since p must vanish in (452) when v = r 3 , we have
9 e -\-4fr 2 -f hr = 0. (456)
From the three last equations may be obtained the values of e, J\
h, in terms of r , V, and JZ; viz.,
. = *^T ^*lgV k= _* (457)
The quantity r , like II and V, is a function of the temperature, the
differential coefficient j — - representing the rate of linear expan-
clt
sion of the solid when without stress.
3 £r.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 375
It will not be necessary to discuss equation (443) at length, as the
case is entirely analogous to that which has just been treated. [It
must be remembered that r/ vi , in the discussion of (443) will take the
place everywhere of the temperature in the discussion of (444).] If
we denote by V and R' the elasticity of volume and the rigidity,
both determined under the condition of constant entropy, (i. e., of no
transmission of heat,) and for states of vanishing stress, we shall
have the equations :
T ~~w+* f ' r * (458)
R' = 2e' + 4f'r 2 , (459)
2 e ' + 4/' r 2 -f h' r — 0. (460)
Whence
In these equations r , R', and V are to be regarded as functions of
the quantity %,.
If we wish to change from one state of reference to another (also
isotropic), the changes required in the fundamental equation are
easily made. If a denotes the length of any line of the solid in the
second state of reference divided by its length in the first, it is evi-
dent that when we change from the first state of reference to the
second the values of the symbols e YI , 7/ v ,, ip Yl , H are divided by a 3 ,
that of E by a 2 , and that of F by a*. In making the change of the
state of reference, we must therefore substitute in the fundamental
equation of the form (444) a 3 ^,, a 2 E, a^F, a 3 JT for ?/>,, E, F,
and H, respectively. In the fundamental equation of the form (443),
we must make the analogous substitutions, and also substitute « 3 7/ v ,
for 7/v,. [It will be remembered that i', e',f, and h' represent func-
tions of 7/v,, and that it is only when their values in terms of 7/ v , are
substituted, that equation (443) becomes a fundamental equation.]
Concerning Solids which absorb Fluids.
There are certain bodies which are solid with respect to some of
their components, while they have other components which are fluid.
In the following discussion, we shall suppose both the solidity and
the fluidity to be perfect, so far as any properties are concerned
which can affect the conditions of equilibrium,—!, e., we shall sup-
pose that the solid matter of the body is entirely free from plasticity,
and that there are no passive resistances to the motion of the fluid
Trans. Conn. Acad., Vol. III. 48 June, 1877.
hX'-i**'. f'- ^Sv ' t t,L-l R '- V ' fa)
376 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
components except such as vanish with the velocity of the motion, —
leaving it to be determined by experiment how far and in what cases
these suppositions are realized.
It is evident that equation (356) must hold true with regard to
such a body, when the quantities of the fluid components contained
in a given element of the solid remain constant. Let FJ, T h \ etc.,
denote the quantities of the several fluid components contained in an
element of the body divided by the volume of the element in the
state of reference, or, in other words, let these symbols denote the
densities which the several fluid components would have, if the body
should be brought to the state of reference while the matter con-
tained in each element remained unchanged. We may then say that
equation (356) will hold true, when rj, F b ! , etc., are constant. The
complete value of the differential of £ v , will therefore be given by an
equation -of the form
,= tdr? v , + 2 2' (x x , £^\ + L a drj + L h dF h ' + etc. (462)
Now when the body is in a state of hydrostatic stress, the term in
this equation containing the signs of summation will reduce to
— pdv v , (vm, denoting, as elsewhere, the volume of the element
divided by its volume in the state of reference). For in this case
de.
/dy dz dz dy\
^'--P \dy'M~dy~'dz~'r
(463)
= — p d
dx dx dx
dx' dy' dz'
dy_ dy_ dy
dx' dy' dz'
dz dz dz
dx' dy' dz'
= —pdv Vl . (464)
We have, therefore, for a state of hydrostatic stress,
d€ y , = t drf Y , - p dv v , + L a drj + L b dr,! + etc., (465)
and multiplying by the volume of the element in the state of refer-
ence, which we may regard as constant,
de=tdi]—pdv + L a dm a + L h dm h -f etc., (466)
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 377
where s, a/, v, m ai m b , etc., denote the energy, entropy, and volume of
the element, and the quantities of its several fluid components. It is
evident that the equation will also hold true, if these symbols are
understood as relating to a homogeneous body of finite size. The
only limitation with respect to the variations is that the element or
body to which the symbols relate shall always contain the same solid
matter. The varied state may be one of hydrostatic stress or
otherwise.
Bat when the body is in a state of hydrostatic stress, and the solid
matter is considered invariable, we have by equation (12)
de = t drf — p dv -j- Ma ^ m a + Mb dm b + etc. (46 1)
It should be remembered that the equation cited occurs in a discus-
sion which relates only to bodies of hydrostatic stress, so that the
varied state as well as the initial is there regarded as one of hydro-
static stress.' But a comparison of the two last equations shows that
the last will hold true without any such limitation, and moreover,
that the quantities X a , L bi etc., when determined for a state of hydro-
static stress, are equal to the potentials jj a , jJ h etc.
Since we have hitherto used the term potential solely with refer-
ence to bodies of hydrostatic stress, we may apply this term as we
choose with regard to other bodies. We may therefore call the quanti-
ties X a , X 6 , etc., the potentials for the several fluid components in the
body considered, whether the state of the body is one of hydrostatic
stress or not, since this use of the term involves only an extension of
its former definition. It will also be convenient to use our ordinary
symbol for a potential to represent these quantities. Equation (462)
may then be written
(dx\
X x , d-=-, \ + /£„ drj 4- Mb drj -f etc. (468)
This equation holds true of solids having fluid components without
any limitation with respect to the initial state or to the variations,
except that the solid matter to which the symbols relate shall remain
the same.
In regard to the conditions of equilibrium for a body of this kind,
it is evident in the first place that if we make X a ', r b \ etc., constant,
we shall obtain from the general criterion of equilibrium all the con-
ditions which we have obtained for ordinary solids, and which are
expressed by the formula3 (364), (374), (380), (382)-(384). The
quantities /'„', X 2 ', etc., in the last two formulae include of course
378 J. W. G-ibbs — Equilibrium of Heterogeneous Substances.
those which have just been represented by rj, F b \ etc., and which
relate to the fluid components of the body, as well as the correspond-
ing quantities relating to its solid components. Again, if we sup-
pose the solid matter of the body to remain without variation in
quantity or position, it will easily appear that the potentials for the
substances which form the fluid components of the solid body must
satisfy the same conditions in the solid body and in the fluids in con-
tact with it, as in the case of entirely fluid masses. See eqs. (22).
The above conditions must however be slightly modified in order
to make them sufficient for equilibrium. It is evident that if the
solid is dissolved at its surface, the fluid components which are set
free may be absorbed by the solid as well as by the fluid mass, and
in like manner if the quantity of the solid is increased, the fluid com-
ponents of the new portion may be taken from the previously exist-
ing solid mass. Hence, whenever the solid components of the solid
body are actual components of the fluid mass, (whether the case is
the same with the fluid components of the solid body or not,) an
equation of the form (383) must be satisfied, in which the potentials
/j a , jjt h etc., contained implicitly in the second member of the equa-
tion are determined from the solid body. Also if the solid compon-
ents of the solid body are all possible but not all actual components
of the fluid mass, a condition of the form (384) must be satisfied, the
values of the potentials in the second member being determined as in
the preceding case.
The quantities
t, AT X ,, . . . Z ZI , yu a , ii h etc., (469)
being differential coefficients of e y , with respect to the variables
*• !>•••§" I":, ZV, etc., (470)
will of course satisfy the necessary relations
dt dX x , ,,.,.
— = -=-i', etc. (471)
,dx drf yt
a M
This result may be generalized as follows. Not only is the second
member of equation (468) a complete differential in its present form,
but it will remain such if we transfer the sign of differentiation (d)
from one factor to the other of any term (the sum indicated by the
symbol -2 2' is here supposed to be expanded into nine terms), and
at the same time change the sign of the term from -f to — . For to
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 379
substitute — r/ v ,dt for tdifr,, for example, is equivalent to subtract-
ing the complete differential d(t rj YI ). Therefore, if we consider the
quantities in (469) and (470) which occur in any same term in equa-
tion (468) as forming a pair, we may choose as independent variables
either quantity of each pair, and the differential coefficient of the
remaining quantity of any pair with respect to the independent
variable of another pair will be equal to the differential coefficient of
the remaining quantity of the second pair with respect to the inde-
pendent variable of the first, taken positively, if the independent
variables of these pairs are both affected by the sign d in equation
(468), or are neither thus affected, but otherwise taken negatively.
Thus
(s?)iHSi- m=-®, «•*
a dx' a dx>
Wjz x , = {dij^: (dFj/xx, = ~ ti)r; ' {473)
where in addition to the quantities indicated by the suffixes, the
following are to be considered as constant: either t or 7/ v , , either
X Y , or — , , . . . either Z z , or yy , either }j b or F b , etc.
It will be observed that when the temperature is constant the con-
ditions jn a = const., /J b = const, represent the physical condition of a
body in contact with a fluid of which the phase does not vary, and
which contains the components to which the potentials relate. Also
that when FJ, r b , etc., are constant, the heat absorbed by the body
in any infinitesimal change of condition per unit of volume measured
in the state of reference is represented by t dr) Y , . If we denote this
quantity by dQ YI , and use the suffix Q to denote the condition of no
transmission of heat, we may write
(d\ogt\ _(dXx\ (d}2%?\ —.( d jk\ (a>ta\
\ J^ JQ ~ \dQvJt' \ dX x , )q- \dQjx x ; (474)
a dx'
jdx
(*3!\-(Jte) ,(*$?)= -(*£l\., (475)
\dXjt \dlogtJx x ,' \/±/t \d\o S t)^,' y '
dx'
where FJ, F b , etc., must be regarded as constant in all the equations,
and either JT Y , or — , , . . . either Z z , or -=-, , in each equation.
cly Cm
380 J. W. Gibbs— Equilibrium of Heterogeneous Substances.
Influence of surfaces of discontinuity upon the equilibrium
of heterogeneous masses. — tlieoky of capillarity.
We have hitherto supposed, in treating of heterogeneous masses in
contact, that they might be considered as separated by mathematical
surfaces, each mass being unaffected by the vicinity of the others, so
that it might be homogeneous quite up to the separating surfaces
both with respect to the density of each of its various components
and also with respect to the densities of energy and entropy. That
such is not rigorously the case is evident from the consideration that
if it were so with respect to the densities of the components it could
not be so in general with respect to the density of energy, as the
sphere of molecular action is not infinitely small. But we know from
observation that it is only within very small distances of such a sur-
face that any mass is sensibly affected by its vicinity, — a natural
consequence of the exceedingly small sphere of sensible molecular
action, — and this fact renders possible a simple method of taking-
account of the variations in the densities of the component substances
and of energy and entropy, which occur in the vicinity of surfaces of
discontinuity. We may use this term, for the sake of brevity, with-
out implying that the discontinuity is absolute, or that the term
distinguishes any surface with mathematical precision. It may be
taken to denote the non-homogeneous film which separates homo-
geneous or nearly homogeneous masses.
Let us consider such a surface of discontinuity in a fluid mass
which is in equilibrium and uninfluenced by gravity. For the pre-
cise measurement of the quantities with which we have to do, it will
be convenient to be able to refer to a geometrical surface, which
shall be sensibly coincident with the physical surface of discontinuity,
but shall have a precisely determined position. For this end, let us
take some point in or very near to the physical surface of discon-
tinuity, and imagine a geometrical surface to pass through this point
and all other points which are similarly situated with respect to the
condition of the adjacent matter. Let this geometrical surface be
called the dividing surface, and designated by the symbol S. It
will be observed that the position of this surface is as yet to a certain
extent arbitrary, but that the directions of its normals are already
everywhere determined, since all the surfaces which can be formed in
the manner described are evidently parallel to one another. Let us
also imagine a closed surface cutting the surface S and including a
part of the homogeneous mass on each side. We will so far limit the
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 381
form of this closed surface as to suppose that on each side of S, as far
as there is any want of perfect homogeneity in the fluid masses, the
closed surface is such as may be generated by a moving normal to S.
Let the portion of S which is included by the closed surface be
denoted by §, and the area of this portion by s. Moreover, let the
mass contained within the closed surface be divided into three parts
by two surfaces, one on each side of S, and very near to that surface,
although at such distance as to lie entirely beyond the influence of
the discontinuity in its vicinity. Let us call the part which contains
the surface s (with the physical surface of discontinuity) M, and the
homogeneous parts M' and M", and distinguish by e, t', a", 77, 7/, 77",
m x , m 3 ', mj*, m 2 , m 2 ', »( 2 ', etc., the energies and entropies of these,
masses, and the quantities which they contain of their various com-
ponents.
It is necessary, however, to define more precisely what is to be
understood in cases like the present by the energy of masses which
are only separated from other masses by imaginary surfaces. A part
of the total energy which belongs to the matter in the vicinity of the
separating surface, relates to pairs of particles which are on different
sides of the surface, and such energy is not in the nature of things
referable to either mass by itself. Yet, to avoid the necessity of
taking separate account of such energy, it will often be convenient to
include it in the energies which we refer to the separate masses.
When there is no break in the homogeneity at the surface, it is
natural to treat the energy as distributed with a uniform density.
This is essentially the case with the initial state of the system which
we are considering, for it has been divided by surfaces passing in
general through homogeneous masses. The only exception — that of
the surface which cuts at right angles the non-homogeneous film —
(apart from the consideration that without any important loss of
generality we may regard the part of this surface within the film as
very small compared with the other surfaces) is rather apparent than
real, as there is no change in the state of the matter in the direction
perpendicular to this surface. But in the variations to be considered
in the state of the system, it will not be convenient to limit ourselves
to such as do not create any discontinuity at the surfaces bounding
the masses M, M', M" : we must therefore determine how we will
estimate the energies of the masses in case of such infinitesimal
discontinuities as may be supposed to arise. Now the energy of
each mass will be most easily estimated by neglecting the discon-
tinuity, i. e., if we estimate the energy on the supposition that
382 J. W. Gibbs— Equilibrium of Heterogeneous Substances.
beyond the bounding surface the phase is identical with that within
the surface. This will evidently be allowable, if it does not affect
the total amount of energy. To show that it does not affect this
quantity, we have only to observe that, if the energy of the mass on
one side of a surface where there is an infinitesimal discontinuity of
phase is greater as determined by this rule than if determined by
any other (suitable) rule, the energy of the mass on the other side
must be less by the same amount when determined by the first rule
than when determined by the second, since the discontinuity relative
to the second mass is equal but opposite in character to the discon-
tinuity relative to the first.
If the entropy of the mass which occupies any one of the spaces
considered is not in the nature of things determined without refer-
ence to the surrounding masses, we may suppose a similar method to
be applied to the estimation of entropy.
With this understanding, let us return to the consideration of the
equilibrium of the three masses M, M', and M". We shall suppose
that there are no limitations to the possible variations of the svstem
due to any want of perfect mobility of the components by means of
which we express the composition of the masses, and that these com-
ponents are independent, i. e., that no one of them can be formed out
of the others.
With regard to the mass M, which includes the surface of discon-
tinuity, it is necessary for its internal equilibrium that when its
boundaries are considered constant, and when we consider only
reversible variations (i. e., those of which the opposite are also
possible), the variation of its energy should vanish with the varia-
tions of its entropy and of the quantities of its various components.
For changes within this mass will not affect the energy or the entropy
of the surrounding masses (when these quantities are estimated on
the principle which we have adopted), and it may therefore be
treated as an isolated system. For fixed boundaries of the mass M,
and for reversible variations, we may therefore write
de = A 6r/ -f A l 6m 1 -f A 2 8m 2 -f etc., (476)
where A , A t , A 2 , etc., are quantities determined by the initial
(unvaried) condition of the system. It is evident that A is the
temperature of the lamelliform mass to which the equation relates,
or the temperature at the surface of discontinuity. By comparison
of this equation with (12) it will be seen that the definition of A Jf
A 2 , etc., is entirely analogous to that of the potentials in homo-
J. W. Gibbs — MquUibrvwm of Heterogeneous Substances. 383
geneous masses, although the mass to which the former quantities
relate is not homogeneous, while in our previous definition of poten-
tials, only homogeneous masses were considered. By a natural ex-
tension of the term potential, we may call the quantities A 1 ,A 2 ,
etc., the potentials at the surface of discontinuity. This designation
will be farther justified by the fact, which will appear hereafter, that
the value of these quantities is independent of the thickness of the
lamina (M) to which they relate. If we employ our ordinary sym-
bols for temperature and potentials, we may write
6s = t drf A- yUj ^j -j- ju 2 6m 2 -\- etc. (4'7'7)
If we substitue ^ for = in this equation, the formula will hold
true of all variations whether reversible or not ;* for if the variation of
energy could have a value less than that of the second member of
the equation, there must be variation in the condition of M in which
its energy is diminished without change of its entropy or of the
quantities of its various components.
It is important, however, to observe that for any given values of
Stj, (Jffij, 8m 2 , etc., while there may be possible variations of the
nature and state of M for which the value of 8s is greater than that
of the second member of (4*77), there must always be possible varia-
tions for which the value of 8 s is equal to that of the second member.
* To illustrate the difference between variations which are reversible, and those
which are not, we may conceive of two entirely different substances meeting in equilib-
rium at a mathematical surface without being at all mixed. We may also conceive of
them as mixed in a thin film about the surface where they meet, and then the amount
of mixture is capable of variation both by increase and by diminution. But when they
are absolutely unmixed, the amount of mixture can be increased, but is incapable of
diminution, and it is then consistent with equilibrium that the value of tie (for a varia-
tion of the system in which the substances commence to mix) should be greater than
the second member of (477). It is not necessary to determine whether precisely such
cases actually occur ; but it would not be legitimate to overlook the possible occur-
rence of cases in which variations may be possible while the opposite variations are
not.
It will be observed that the sense in which the term reversible is here used is en-
tirely different from that in which it is frequently used in treatises on thermody-
namics, where a process by which a system is brought from a state A to a state B is
called reversible, to signify that the system may also be brought from the state B to
the state A through the same series of intermediate states taken in the reverse order
by means of external agencies of the opposite character. The variation of a system
from a state A to a state B (supposed to differ infinitely little from the first) is here
called reversible when the system is capable of another state B' which bears the same
relation to the state A that A bears to B.
Teans. Conn. Acad., Vol. III. 49 June, 1877.
384 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
It will be convenient to have a notation which will enable us to ex-
press this by an equation. Let bf denote the smallest value (i. e., the
value nearest to — go) of 8 s consistent with given values of the
other variations, then
be = t 8rj -|- ju l 8m 1 -{• pi 2 6m 2 -j- etc. (478)
For the internal equilibrium of the whole mass which consists of
the parts M, M', M", it is necessary that
d € _j_ Se' -f 8e" ^ (479)
for all variations which do not affect the enclosing surface or the
total entropy or the total quantity of any of the various components.
If we also regard the surfaces separating M, M', and M" as invaria-
ble, we may derive from this condition, by equations (478) and (12),
the following as a necessary condition of equilibrium :
t di] -f- Mi $ m i + M2 8 m 2 + etc -
-f- t' drf -f- jd t ' 8m/ -\- ja 2 ' 8m 2 -f- etc.
-f t" 8rj" + /jl x " 8m/' + M 2 " 8m z " + etc. i 0, (480)
the variations being subject to the equations of conditions
dri + 8rf + 8rf = 0, ^)
*,», + **<+*!»/ = <), I (4gi)
8m 2 -f- Sm 2 ' -j- Sm 2 " = 0,
etc.
It may also be the case that some of the quantities 6m/, 6m/,
6m 2 ', 8m 2 ", etc., are incapable of negative values or can only have
the value zero. This will be the case when the substances to which
these quantities relate are not actual or possible components of M'
or M". (Seepage 117.) To satisfy the above condition it is neces-
sary and sufficient that
t = t' = f, (482)
^i'^i' = i"] tfm i') Ma dm 2 '^jA 2 8m 2 , etc., (483)
ti/'6m 1 w ^ji x 8m x \ /V' ( W = J M 2 ( W 5 etc. (484)
It will be observed that, if the substance to which //,, for instance,
relates is an actual component of each of the homogeneous masses,
we shall have //, = m/ = ///. If it is an actual component of the
first only of these masses, we shall have jj 1 = ju/. If it is also a
possible component of the second homogeneous mass, we shall also
have jx 1 =jj- 1 ". If this substance occurs only at the surface of dis-
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 385
continuity, the value of the potential yu, will not be determined by
any equation, but cannot be greater than the potential for the same
substance in either of the homogeneous masses in which it may be a
possible component.
It appears, therefore, that the particular conditions of equilibrium
relating to temperature and the potentials which we have before
obtained by neglecting the influence of the surfaces of discontinuity
(pp. 119, 120, 128) are not invalidated by the influence of such dis-
continuity in their application to homogeneous parts of the system
bounded like M' and M" by imaginary surfaces lying within the
limits of homogeneity, — a condition which may be fulfilled by sur-
faces very near to the surfaces of discontinuity. It appears also that
similar conditions will apply to the non-homogeneous films like M',
which separate such homogeneous masses. The properties of such
films, which are of course different from those of homogeneous
masses, require our farther attention.
The volume occupied by the mass M is divided by the surface s
into two parts, which we will call v'" and v'"\ v'" lying next to M^
and v"" to M". Let us imagine these volumes filled by masses hav-
ing throughout the same temperature, pressure and potentials, and
the same densities of energy and entropy, and of the various com-
ponents, as the masses M' and M" respectively. We shall then have,
by equation (12), if we regard the volumes as constant,
oV" = t' 67f + yu/ 6m 1 I " + }a 2 ' 6m 2 " ! -f- etc., (485)
6s"" = t" 6?f" + Mi" <W" + H" <W" + etc. ; (486)
whence, by (482)-(484), we have for reversible variations
6s'" = t 6rf" + /j 1 6m ^ + ju 8 6m 2 '" -\- etc., (487)
6s"" — tdi]"" -f- Mi om/"' + yu 2 6m 2 "" + etc. (488)
From these equations and (4*77), we have for reversible variations
S( £ _ e ! " - a"") = t 6{ V - if - ?f")
+ jj x ^(m, - j»,'" - Mj"") + jj 2 d \ m 2 ~ m 2'" ~ m s"") -fete. (489)
Or, if we set*
£ s _ s _ e "> _ f» 9 jf - v - if - if", (490)
m\ =. m l — m"' — w./'", m\ = m 2 — m 2 '" — m 2 "", etc., (491)
* It will be understood that the s here used is not an algebraic exponent, but is
only intended as a distinguishing mark. The Koman letter S has not been used to
denote any quantity.
386 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
we may write
de s = t d?f -f ja | Sm\ + // 2 6m\ -f etc. (492)
This is true of reversible variations in which the surfaces which have
been considered are fixed. It will be observed that t s denotes the
excess of the energy of the actual mass which occupies the total
volume which we have considered over that energy which it would
have, if on each side of the surface S the density of energy had the
same uniform value quite up to that surface which it has at a sensi-
ble distance from it; and that rf, m^, m|, etc, have analogous signi-
fications. It will be convenient, and need not be a source of any
misconception, to call £ s and rf the energy and entropy of the surface
(or the superficial energy and entropy), — and — the superficial den-
s s
711/ 771
sities of energy and entropy, — -, — -, etc., the superficial densities of
the several components.
Now these quantities (e s , if, m\, etc.) are determined partly by
the state of the physical system which we are considering, and partly
by the various imaginary surfaces by means of which these quanti-
ties have been defined. The position of these surfaces, it will be
remembered, has been regarded as fixed in the variation of the sys-
tem. It is evident, however, that the form of that portion of these
surfaces, which lies in the region of homogeneity on either side of the
surface of discontinuity cannot affect the values of these quantities.
To obtain the complete value of de s for reversible variations, we have
therefore only to regard variations in the position and form of the
limited surface s, as this determines all of the surfaces in question
lying within the region of non-homogeneity. Let us first suppose
the form of s to remain unvaried and only its position in space to
vary, either by translation or rotation. No change in (492) will be
necessary to make it valid in this case. For the equation is valid if
8 remains fixed and the material system is varied in position ; also, if
the material system and § are both varied in position, while their
relative position remains unchanged. Therefore, it will be valid if
the surface alone varies its position.
But if the form of s be varied, we must add to the second member
(492) terms which shall represent the value of
6V — t 6rf — // , 8m\ — }a 2 dm\ — etc.
due to such variation in the form of §, If we suppose s to be suffi-
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 387
ciently small to be considered uniform throughout in its curvatures
and in respect to the state of the surrounding matter, the value of the
above expression will be determined by the variation of its area ds
and the variations of its principal curvatures 6c x and 6c 2 , and we
may write
6e s ■=. t 61 f -f Mi ^ w i +7^2 ^ m \ + etc -
+ <5 ds + C\ 6c, + C\ 6c 2 , (493)
or
6s s — t 6?f -\- fj. 1 6m\ -f- jj 2 6m\ + etc.
+ ads + HC, + C 2 ) 6{c, + c 2 ) -fi(6\ - G 2 ) 6( Cl - c 2 ),(494)
c, 6\, and C 3 denoting quantities which are determined by the
initial state of the system and position and form of g. The above is
the complete value of the variation of £ s for reversible variations of
the system. But it is always possible to give such a position to the
surface g that O x -f C 2 shall vanish.
To show this, it will be convenient to write the equation in the
longer form [see (490), (491)]
6e — t 6rj — /a , 6m l — yu 2 6m 2 — etc.
__ Se'" + t 67 f + fi 1 6m 1 '" + }a 2 dm J" + etc.
_ 6Y"' + t d V "" ~f Ml 6m,"" + // 2 6m 2 "" + etc.
= o- 6s + i (6\ + tf a ) d(fl a + c 2 ) + i {C t - C 2 ) 6( Cl ~c 2 ), (495)
i. e., by (482)-(484) and (12),
Se - t 6r) - /i, Sm l — ,u 2 om 2 - etc. +y 6V" +£>" 6V'"
= a" <fe + i (C, + (?,) 6( Cl + c a ) + * ( C\ - <7 2 ) *(<?, - c 2 ). (496)
From this equation it appears in the first place that the pressure is
the same in the two homogeneous masses separated by a plane sur-
face of discontinuity. For let us imagine the material system to
remain unchanged, while the plane surface g without change of area
or of form moves in the direction of its normal. As this does not
affect the boundaries of the mass M,
6e — t 6rj — jjl x 6m t — jj 2 6m 2 — etc. = 0.
Also 6s = 0, 6(c 1 -\-c 2 ) = 0, 6(c t - c 2 ) = 0, and 6v'" ~ - 6v"".
Hence p' = p" , when the surface of discontinuity is plane.
Let us now examine the effect of different positions of the surface g
in the same material system upon the value of C, -j- 21 supposing at
first that in the initial state of the system the surface of discontinuity
is plane. Let us give the surface g some particular position. In the
388 J. W. Gibbs — Equilibrium of Heterogeneous Substances,
initial state of the system this surface will of course be plane like the
physical surface of discontinuity, to which it is parallel. In the
varied state of the system, let it become a portion of a spherical
surface having positive curvature ; and at sensible distances from
this surface let the matter be homogeneous and with the same phases
as in the initial state of the system ; also at and about the surface let
the state of the matter so far as possible be the same as at and about
the plane surface in the initial state of the system. (Such a variation
in the system may evidently take place negatively as well as posi-
tively, as the surface may be curved toward either side. But
whether such a variation is consistent with the maintenance of equi-
librium is of no consequence, since in the preceding equations only
the initial state is supposed to be one of equilibrium.) Let the
surface s, placed as supposed, whether in the initial or the varied
state of the surface, be distinguished by the symbol s'. Without
changing either the initial or the varied state of the material system,
let us make another supposition with respect to the imaginary sur-
face s. In the unvaried system let it be parallel to its former posi-
tion but removed from it a distance A on the side on which lie the
centers of positive curvature. In the varied state of the system, let
it be spherical and concentric with s', and separated from it by the
same distance A. It will of course lie on the same side of s' as in the
unvaried system. Let the surface s, placed in accordance with this
second supposition, be distinguished by the symbol s". Both in the
initial and the varied state, let the perimeters of §' and s" be traced
by a common normal. Now the value of
6s — t drj — yUj tfm, — pi 2 Sm 2 — etc.
in equation (496) is not affected by the position of s, being deter-
mined simply by the body M : the same is true p' 6V" + p" 8v"" or
p'S(v'" + v""),-v'"-\- v"" being the volume of M. Therefore the second
member of (496) will have the same value whether the expressions
relate to s' or §". Moreover, S(c 1 — c 2 ) = both for s' and s". If
we distinguish the quantities determined for s' and for s" by the
marks ' and ", we may therefore write
o-'oY+|(C7+ G 2 ') S( Cl ' + c 2 ') = 0"Ss" + i(C/+ C 2 ")6( Cl " + c 2 ").
Now if we make ds" = 0,
we shall have by geometrical necessity
dYzTsAd^c/'+c/).
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 389
Hence
o-'s\d(cJ + c/) + %(G\'+CJ)d(c 1 ' + cJ)=UCJ'+CJ')d(oJ' + eJ').
But d(c l '-\-cJ) = d(c 1 "+c 2 ").
Therefore, CJ+ CJ + 2 &' s X = C\" + C 2 ".
This equation shows that we may give a positive or negative value
to CJ + C 2 " by placing s" a sufficient distance on one or on the
other side of §'. Since this is true when the (unvaried) surface is
plane, it must also be true when the surface is nearly plane. And for
this purpose a surface may be regarded as nearly plane, when the
radii of curvature are very large in proportion to the thickness of the
non-homogeneous film. This is the case when the radii of curvature
have any sensible size. In general, therefore, whether the surface of
discontinuity is plane or curved it is possible to place the surface s
so that C 1 + C 2 in equation (494) shall vanish.
Now we may easily convince ourselves by equation (493) that if s
is placed within the non-homogeneous film, and s== 1, the quantity 6
is of the same order of magnitude as the values of £ s , ?f, m\, m|, etc.,
while the values of C l and C 2 are of the same order of magnitude
as the changes in the values of the former quantities caused by
increasing the curvature of s by unity. Hence, on account of the
thinness of the non-homogeneous film, since it can be very little
aifected by such a change of curvature in s, the values of C, and G 2
must in general be very small relatively to o'. And hence, if s' be
placed within the non-homogeneous film, the value of A which will
make C J + C.J' vanish must be very small (of the same order of
magnitude as the thickness of the non-homogeneous film). The posi-
tion of s, therefore, which will make C x + C 2 in (494) vanish, will
in general be sensibly coincident with the physical surface of
discontinuity.
We shall hereafter suppose, when the contrary is not distinctly
indicated that the surface s, in the unvaried state of the system, has
such a position as to make C 1 + C 2 =. 0. It will be remembered that
the surface s is a part of a larger surface S, which we have called the
dividing surface, and which is coextensive with the physical surface
of discontinuity. We may suppose that the position of the dividing
surface is everywhere determined by similar considerations. This
is evidently consistent with the suppositions made on page 380 with
regard to this surface.
390 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
We may therefore cancel the term
|((7 1 +a 2 )6> 1 + C2 )
in (494). In regard to the following term, it will be observed that
G t must necessarily be equal to C\, when c 1 = c a , which is the case
when the surface of discontinuity is plane. Now on account of the
thinness of the non-homogeneous film, we may always regard it as
composed of parts which are approximately plane. Therefore, with-
out danger of sensible error, we may also cancel the term
Equation (494) is thus reduced to the form
oV = t 3r/ & + o- 6s+/i 1 6m\ + /j 2 dm% -f etc. (497)
We may regard this as the complete value of Se s , for all reversible
variations in the state of the system supposed initially in equilibrium,
when the dividing surface has its initial position determined in the
manner described.
The above equation is of fundamental importance in the theory
of capillarity. It expresses a relation with regard to surfaces of dis-
continuity analogous to that expressed by equation (12) with regard
to homogeneous masses. From the two equations may be directly
deduced the conditions of equilibrium of heterogeneous masses in con-
tact, subject or not to the action of gravity, without disregard of the
influence of the surfaces of discontinuity. The general problem, in-
cluding the action of gravity, we shall take up hereafter: at present
we shall only consider, as hitherto, a small part of a surface of dis-
continuity with a part of the homogeneous mass on either side, in
order to deduce the additional condition which may be found when
we take account of the motion of the dividing surface.
We suppose as before that the mass especially considered is
bounded by a surface of which all that lies in the region of non-
homogeneity is such as may be traced by a moving normal to the
dividing surface. But instead of dividing the mass as before into
four parts, it will be sufficient to regard it as divided into two parts
by the dividing surface. The energy, entropy, etc., of these parts,
estimated on the supposition that its nature (including density of
energy, etc.) is uniform quite up to the dividing surface, will be
denoted by s', rj , etc., e", if, etc. Then the total energy will be
s s + €'+e", and the general condition of internal equilibrium will be
that
<?£ s +oY+dV^0, (498)
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 391
when the bounding surface is fixed, and the total entropy and total
quantities of the various components are constant. We may suppose
if, if, if, m\, m,', m/, m%, m 2 ', m 2 ", etc., to be all constant. Then
by (497) and (12) the condition reduces to
o' 6s - p oV - p" oV = 0. (499)
(We may set = for ^, since changes in the position of the dividing
surface can evidently take place in either of two opposite directions.)
This equation has evidently tlie same form as if a membrane without
rigidity and having a tension o~, uniform in all directions, existed
at the dividing surface. Hence, the particular position which we
have chosen for this surface may be called the surface of tension, and
a the superficial tension. If all parts of the dividing surface move
a uniform normal distance 6JV, we shall have
6s = (<y 1 +c 2 ) s SN, oV = s 61V, 6v" = - s 6N;
whence ff (c, + c 2 ) =p' — p\ (500)
the curvatures being positive when their centers lie on the side to
which/>' relates. This is the condition which takes the place of that
of equality of pressure (see pp. 119, 128) for heterogeneous fluid
masses in contact, when we take account of the influence of the sur-
faces of discontinuity. We have already seen that the conditions
relating to temperature and the potentials are not affected by these
surfaces.
Fundamental Equations for Surfaces of Discontinuity. @j2$-
In equation (497) the initial state of the system is supposed to be
one of equilibrium. The only limitation with respect to the varied
state is that the variation shall be reversible, i. e., that an opposite
variation shall be possible. Let us now confine our attention to
variations in which the system remains in equilibrium. To distin-
guish this case, we may use the character d instead 6, and write
de s = t drf + 6 ds-\- fi 1 dm\ + /u 2 dm% + etc. (501)
Both the states considered being states of equilibrium, the limitation
with respect to the reversibility of the variations may be neglected,
since the variations will always be reversible in at least one of the
states considered.
If we integrate this equation, supposing the area s to increase
from zero to any finite value 6', while the material system to a part
of which the equation relates remains without change, we obtain
6 s = t if -f- G s + fi j m s , 4- )J 2 m\ + etc., (502)
Trans, Conn. Acad., Vol. III. 50 July, 1877.
r x = -i, F 2 = -*, etc., (505)
392 J. W. Gribbs —Hl'iuilibrium of Heterogeneous Substances.
which may be applied to any portion of any surface of discontinuity
(in equilibrium) which is of the same nature throughout, or through-
out which the values of t, ff, yu,, /u 2 , etc. are constant.
If we differentiate this equation, regarding all- the quantities as
variable, and compare the result with (501), we obtain
if dt + s da -f m\ dfi 1 + m\ dju 2 + etc. = 0. (503)
If we denote the superficial densities of energy, of entropy, and
of the several component substances (see page 386) by s s , r/ s , XT,, F 2 ,
etc., we have
£ s s
8 S =-, *h = -j, (504)
s ' * s
and the preceding equations may be reduced to the form : —
ds s = t dr/ s + yu, dT x + M2 dF 2 + etc., (506)
e 8 .= t % -f + M i r % 4- M 2 r 2 + etc -> (»0?)
eftr =z — 7/ s dt — J 7 , (iyMj — T 2 d/u 2 — etc. (508)
Now the contact of the two homogeneous masses does not impose
any restriction upon the variations of phase of either, except that
the temperature and the potentials for actual components shall have
the same value in both. [See (482)-(484) and (500).] For however
the values of the pressures in the homogeneous masses may vary (on
account of arbitrary variations of the temperature and potentials),
and however the superficial tension may vary, equation (500) may
always be satisfied by giving the proper curvature to the surface of
tension, so long, at least, as the difference of pressures is not great.
Moreover, if any of the potentials ja 1 , /u 2 , etc. relate to substances
which are found only at the surface of discontinuity, their values
may be varied by varying the superficial densities of those sub-
stances. The values of t, yw 15 /* s , etc. are therefore independently
variable, and it appears from equation (508) that 6 is a function of
these quantities. If the form of this function is known, we may
derive from it by differentiation w + 1 equations (n denoting the total
number of component substances) giving the values of %, r x ,'F 2 ,
etc. in terras of the variables just mentioned. This will giye us,
with (507), w+ 3 independent equations between the 2?? + 4 quantities
which occur in that equation. These are all that exist, since n+ 1
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 393
of these quantities are independently variable. Or, we may consider
that we have n+3 independent equations between the 2/1 + 5 quan-
tities occurring in equation (5i»2), of which n+2 are independently
variable.
An equation, therefore, between
<?, t, Mi, M 2 , etc -> ( 509 )
may be called a fundamental equation for the surface of discontinuity.
An equation between
e s , ?f, s, m\ m|, etc., (510)
or between s s , ?/ s , F x , i" 2 , etc., (5H)
may also be called a fundamental equation in the same sense. For
it is evident from (501) that an equation may be regarded as subsist-
ing between the variables (510), and if this equation be known, since
n -f- 2 of the variables may be regarded as independent (viz., n -\- 1
for the n -\- 1 variations in the nature of the surface of discontinuity,
and one for the area of the surface considered), we may obtain by
differentiation and comparison with (501), n -f- 2 additional equations
between the 2n -\- 5 quantities occurring in (502). Equation (506)
shows that equivalent relations can be deduced from an equation
between the variables (511). It is moreover quite evident that an
equation between the variables (510) must be reducible to the form
of an equation between the ratios of these variables, and therefore to
an equation between the variables (511).
The same designation may be applied to any equation from which,
by differentiation and the aid only of general principles and relations,
n + 3 independent relations between the same In -j- 5 quantities
may be obtained.
If we set ip s = £ s /5 - trf, (512)
we obtain by differentiation and comparison with (501)
dip s = — if dt -\- 6 ds -j- fi 1 dm\ -f fx 2 dm\ -\- etc. (513)
An equation, therefore, between ip s , t, s, m s n m|, etc., is a fundamental
equation, and is to be regarded as entirely equivalent to either of the
other fundamental equations which have been mentioned.
The reader will not fail to notice the analogy between these funda-
mental equations, which relate to surfaces of discontinuity, and those
relating to homogeneous masses, which have been described on pages
140-144.
3 94 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
On the Experimental Determination of Fundamental Equations for
Surfaces of Discontinuity.
When all the substances which are found at a surface of discon-
tinuity are components of one or the other of the homogeneous
masses, the potentials w 15 jj 2 , etc., as well as the temperature, may
be determined from these homogeneous masses.* The tension o' may
be determined by means of the relation (500). But our measure-
ments are practically confined to cases in which the difference of the
pressures in the homogeneous masses is small ; for with increasing
differences of pressure the radii of curvature soon become too small
for measurement. Therefore, although the equation p' z=zp" (which
is equivalent to an equation between t, /*,, jj 2 , etc., since p' and p"
are both functions of these variables) may not be exactly satisfied in
cases in which it is convenient to measure the tension, yet this equa-
tion is so nearly satisfied in all the measurements of tension which
we can make, that we must regard such measurements as simply
establishing the values of a for values of t, /u t , /v 2 , etc., which satisfy
the equation p' = p", but not as sufficient to establish the rate of
change in the value of 6 for variations of t, pt 1} // 2 , etc., which are
inconsistent with the equation jt?' —p".
To show this more distinctly, let t, yu 2 , m 3 , etc. remain constant,
then by (508) and (98)
do~ = — r i d/* 1 ,
dp' = y 1 'd/* 1 ,
dp" = y 1 "d/i 1 ,
y ± ' and y x " denoting the densities — f and — ~. Hence,
dp'-dp"=( ri '- ri ")d Ml ,
and r, d(p' - p") = {y ," - y t ') do:
But by (500)
( c i + G 2 ) d>G + tf ^{ G i + G 2) — d(p'—p").
Therefore,
r i (o a + c 2 ) da + T x 6 d(c ± + c 2 ) — {y x " - y t ') dcr,
or \Yi ' — Yy - r i (c, + c 2 )\dff = T l ed(c 1 + c 8 ).
* It is here supposed that the thermodynamic properties of the homogeneous
masses have already been investigated, and that the fundamental equations of these
masses may be regarded as known.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 395
Now I\ (c 1 + c 2 ) will generally be very small compared with
y i —y x* Neglecting the former term, we have
To integrate this equation, we may regard T x , ;/,', y x " as constant.
This will give, as an approximate value,
a' denoting the value of a when the surface is plane. From this it
appears that when the radii of curvature have any sensible magni-
tude, the value of o' will be sensibly the same as when the surface is
plane and the temperature and all the potentials except one have
the same values, unless the component for which the potential has
not the same value has very nearly the same density in the two
homogeneous masses, in which case, the condition under which the
variations take place is nearly equivalent to the condition that the
pressures shall remain equal.
Accordingly, we cannot in general expect to determine the superfi-
(da \ *
— — 1 by measurements of super-
ficial tensions. The case will be the same with F 2 , r 3 , etc., and also
with 7/ s , the superficial density of entropy.
The quantities f s , //§, r it F 2 , etc. are evidently too small in general
to admit of direct measurement. When one of the components,
however, is found only at the surface of discontinuity, it may be
more easy to measure its superficial density than its potential. But
except in this case, which is of secondary interest, it will generally
be easy to determine <7 in terms of t, yu 15 yu 2 , etc., with considerable
accuracy for plane surfaces, and extremely difficult or impossible to
determine the fundamental equation more completely.
Fundamental Equations for Plane Surfaces of Discontinuity.
An equation giving a in terms of t, ju t , ju 2 , etc., which will hold
true only so long as the surface of discontinuity is plane, may be
called a fundamental equation for a plane surface of discontinuity.
It will be interesting to see precisely what results can be obtained from
such an equation, especially with respect to the energy and entropy
* The suffixed /z is used to denote that all the potentials except that occurring in
the denominator of the differential coefficient are to be regarded as constant.
396 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
and the quantities of the component substances in the vicinity of the
surface of discontinuity.
These results can be exhibited in a more simple form, if we deviate
to a certain extent from the method which we have been following.
The particular position adopted for the dividing surface (which
determines the superficial densities) was chosen in order to make the
term £ ( G x -\- C 2 ) d (e 1 -f c 2 ) in (494) vanish. But when the curvature
of the surface is not supposed to vary, such a position of the divid-
ing surface is not necessary for the simplification of the formula. It
is evident that equation (501) will hold true for plane surfaces (sup-
posed to remain such) without reference to the position of the divid-
ing surfaces, except that it shall be parallel to the surface of discon-
tinuity. We are therefore at liberty to choose such a position for
the dividing surface as may for any purpose be convenient.
None of the equations (502)-(513), which are either derived from
(501), or serve to define new symbols, will be affected by such a
change in the position of the dividing surface. But the expressions
£ s , t/ s , m\, ra|, etc., as also f s , r/ s , r„ f 2 , etc. and tp s , will of course
have different values when the position of that surface is changed.
The quantity <7, however, which we may regard as defined by equa-
tions (501), or, if we choose, by (502) or (507), will not be affected in
value by such a chauge. For if the dividing surface be moved a
distance A. measured normally and toward the side to which v" relates,
the quantities
£ s? t/s) r x , i 2 , etc.,
will evidently receive the respective increments
A(£ V "-£ V '), \ {Vy " - W '), Mx/-r/)» Mr/ -72'), etc.,
8 y \ £ v ", r/y, r/ v " denoting the densities of energy and entropy in the
two homogeneous masses. Hence, by equation (507), 6 will receive
the increment
x(s Y "-e v ')-t\(i ?v "—T7 V ')-n 1 \{y 1 "— ri'H/^Mr/- r/) - etc -
But by (93)
- p" = e v " - t r, v " - /<! y x " - m 2 Yz" ~ etc -,
- p' — Ey - t V - M-i Y\ - M2 V* ~ etc -
Therefore, since j>'=p", the increment in the value of a is zero.
The value of 0' is therefore independent of the position* of the divid-
ing surface, when this surface is plane. But when we call this quan-
tity the superficial tension, we must remember that it will not have
J. W. Gibbs — ^Equilibrium of Heterogeneous Substances. 397
its characteristic properties as a tension with reference to any arbi-
trary surface. Considered as a tension, its position is in the surface
which we have called the surface of tension, and, strictly speaking,
nowhere else. The positions of the dividing surface, however, which
we shall consider, will not vary from the surface of tension sufficiently
to make this distinction of any practical importance.
It is generally possible to place the dividing surface so that the
total quantity of any desired component in the vicinity of the surface
of discontinuity shall be the same as if the density of that component
were uniform on each side quite up to the dividing surface. In other
words, we may place the dividing surface so as to make any one of
the quantities I\, F 2 , etc., vanish. The only exception is with
regard to a component which has the same density in the two homo-
geneous masses. With regard to a component which has very nearly
the same density in the two masses such a location of the dividing
surface might be objectionable, as the dividing surface might fail to
coincide sensibly with the physical surface of discontinuity. Let us
suppose that y x ' is not equal (nor very nearly equal) to y x ", and that
the dividing surface is so placed as to make 7\ = 0. Then equation
(508) reduces to
dff = — ?fa u dt — r 2(1) dju 2 - r 3U) d/x 3 — etc., (514)
where the symbols // S(1) , -T a(1) , etc., are used for greater distinctness
to denote the values of 7/ s , T 2 , etc., as determined by a dividing sur-
face placed so that F 1 = 0. Now we may consider all the differen-
tials in the second member of this equation as independent, without
violating the condition that the surface shall remain plane, i. e., that
dp' =. dp". This appears at once from the values of dp' and dp"
given by equation (98). Moreover, as has already been observed,
when the fundamental equations of the two homogeneous masses are
known, the equation^' =p" affords a relation between the quantities
t, /i 1} yu 2 , etc. Hence, when the value of a is also known for plane
surfaces in terms of t, ju x , p 2 , etc., we can eliminate yUj from this ex-
pression by means of the relation derived from the equality of pres-
sures, and obtain the value of <T for plane surfaces in terms of
t, ju 2 , yu 3 , etc. From this, by differentiation, we may obtain directly
the values of 7/ S(1) , r 2Cl)? F 3(1) , etc., in terms of t, /a 2 , /a 3 , etc. This
would be a convenient form of the fundamental equation. But, if the
elimination of _p',j/, and ja x from the finite equations presents alge-
braic difficulties, we can in all cases easily eliminate dp', dp", dpt x
from the corresponding differential equations and thus obtain a
r 2(l) —
398 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
differential equation from which the values of ^/ S(] ), r. 2ii) , ^' 3 (i)>
etc. in terms of t, yu n pi 2 , etc., may be at once obtained by comparison
with (514).*
* If liquid mercury meets the mixed vapors of water and mercury in a plane sur-
face, and we use /x l and /a 2 to denote the potentials of mercury and water respec-
tively, and place the dividing surface so that r, =0, i. e., so that the total quantity of
mercury is the same as if the liquid mercury reached this surface on one side and the
mercury vapor on the other without change of density on either side, then r 2 (D will
represent the amount of water in the vicinity of this surface, per unit of surface,
above that which there would be, if the water-vapor just reached the surface without
change of density, and this quantity (which we may call the quantity of water con-
densed upon the surface of the mercury) will be determined by the equation
da
' dfi. 2 -
(In this differential coefficient as well as the following, the temperature is supposed
to remain constant and the surface of discontinuity plane. Practically, the latter con-
dition may be regarded as fulfilled in the case of any ordinary curvatures.)
If the pressure in the mixed vapors conforms to the law of Dalton (see pp. 215, 218),
we shall have for constant temperature
dp-2 = J-i dfi. 2 ,
where p 2 denotes the part of the pressure in the vapor due to the water- vapor, and
y 2 the density of the water-vapor. Hence we obtain
da
r 2 a> = -7«^7-
For temperatures below 100° centigrade, this will certainly be accurate, since the pres-
sure due to the vapor of mercury may be neglected.
The value of a forp 2 =0 and the temperature of 20° centigrade must be nearly the
same as the superficial tension of mercury in contact with air, or 55.03 grammes per
linear metre according to Quincke (Pogg. Ann., Bd. 139, p. 27). The value of a at the
same temperature, when the condensed water begins to have the properties of water
in mass, will be equal to the sum of the superficial tensions of mercury in contact
with water and of water in contact with its own vapor. This will be, according to
the same authority, 42.58 + 8.25, or 50.83 grammes per metre, if we neglect the differ-
ence of the tensions of water with its vapor and water with air. As p 2 , therefore,
increases from zero to 236400 grammes per square metre (when water begins to be
condensed in mass), a diminishes from about 55.03 to about 50.83 grammes per linear
metre. If the general course of the values of a for intermediate values of p 2 were
determined by experiment, we could easily form an approximate estimate of the
values of the superficial density r 3(x) for different pressures less than that of satu-
rated vapor. It will be observed that the determination of the superficial density
does not by any means depend upon inappreciable differences of superficial tension.
The greatest difficulty in the determination would doubtless be that of distinguishing
between the diminution of superficial tension due to the water and that due to other
substances which might accidentally be present. Such determinations are of con-
siderable practical importance on account of the use of mercury in measurements of
the specific gravity of vapors.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 399
The. same physical relations may of course be deduced without
giving up the use of the surface of tension as a dividing surface, but
the formulae which express them will be less simple. If we make
t, /i 3 , // 4 , etc. constant, we have by (98) and (508)
dp' = y t r d/j. 1 + y 2 d/.t 2 ,
dp" =y 1 " dp x + y/ d/J 2 ,
dff = — r i d}A 1 —T 2 d/d 2 ,
where we may suppose r x and F 2 to be determined with reference
to the surface of tension. Then, if dp' =dp",
0V ~ Yi") d / J i + 0V - Yz") d M 2 = °,
and
That is,
dff =. I\ I- 2 -, ¥±. H d}A 2 — F 2 djj 2 .
dff '
-) =~ J \ + r t ^4—^- (515)
dM 2 / p'-p",t :F ,,^, etc. Yi-Yx
The reader will observe that — 7 — - — Tf represents the distance be-
tween the surface of tension and that dividing surface which would
make F x = ; the second number of the last equation is therefore
equivalent to —r %(x) .
If any component substance has the same density in the two homo-
geneous masses separated by a plane surface of discontinuity, the
value of the superficial density for that component is independent
of the position of the dividing surface. In this case alone we may
derive the value of the superficial density of a component with
reference to the surface of tension from the fundamental equation for
plane surfaces alone. Thus in the last equation, when y 2 ' = y 2 ", the
second member will reduce to — T 2 . It will be observed that to
make p' ' -~-p", t, yu 3 , /i 4 , etc. constant is in this case equivalent to
making t, yu 1? // 3 , /* 4 , etc. constant.
Substantially the same is true of the superficial density of entropy
or of energy, when either of these has the same density in the two
homogeneous masses.*
* With respect to questions which concern only the form of surfaces of discontinuity,
such precision as we have employed in regard to the position of the dividing surface
is evidently quite unnecessary. This precision has not been used for the sake of the
mechanical part of the problem, which does not require the surface to be defined
with greater nicety than we can employ in our observations, but in order to give
Teans. Conn. Acad., Vol. III. 51 July, 18*77.
400 J. W. Gibbs—Equilibriwn of Heterogeneous Substances.
Concerning the Stability of Surfaces of Discontinuity.
We shall first consider the stability of a film separating homoge-
neous masses with respect to changes in its nature, while its position
and the nature of the homogeneous masses are not altered. For this
purpose, it will be convenient to suppose that the homogeneous
masses are very large, and thoroughly stable with respect to the
possible formation of any different homogeneous masses out of their
components, and that the surface of discontinuity is plane and
uniform.
Let us distinguish the quantities which relate to the actual com-
ponents of one or both of the homogeneous masses by the suffixes
ffi , 5 , etc., and those which relate to components which are found only
at the surface of discontinuity by the suffixes ? , h , etc., and consider
the variation of the energy of the whole system in consequence of a
given change in the nature of a small part of the surface of discon-
tinuity, while the entropy of the whole system and the total quan-
tities of the several components remain constant, as well as the
volume of each of the homogeneous masses, as determined by the
surface of tension. This small part of the surface of discontinuity in
its changed state is supposed to be still uniform in nature, and such
as may subsist in equilibrium between the given homogeneous
masses, which will evidently not be sensibly altered in nature or ther-
modynamic state. The remainder of the surface of discontinuity is
also supposed to remain uniform, and on account of its infinitely greater
size to be infinitely less altered in its nature than the first part. Let
As s denote the increment of the superficial energy of this first part,
Ar/ S , Ami , Almf, etc., Am*, Amf, etc., the increments of its superficial
determinate values to the superficial densities of energy, entropy, and the component
substances, which quantities, as has been seen, play an important part in the relations
between the tension of a surface of discontinuity, and the composition of the masses
which it separates.
The product a s of the superficial tension and the area of the surface, may be
regarded as the available energy due to the surface in a system in which the tempera-
ture and the potentials //,, //.,, etc. — or the differences of these potentials and the
gravitational potential (see page 208) when the system is subject to gravity— are
maintained sensibly constant. The value of a, as well as that of s, is sensibly inde-
pendent nf the precise position which we may assign to the dividing surface (so long
as this is sensibly coincident with the surface of discontinuity), but £ S , the superficial
density of energy, as the term is used in this paper, like the superficial densities of
entropy and of the component substances, requires a more precise localization of the
dividing surface.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 401
entropy and of the quantities of the components which we regard
as belonging to the surface. The increments of entropy and of the
various components which the rest of the system receive will be
expressed by
— Aif, — Am*, — Ami, etc., — 4 m*, —Am*, etc.,
and the consequent increment of energy will be by (12) and (501)
— t Arf — jj a Ami — Vb Alm\ — etc. — jj g Am* — ji 1t Am\ — etc.
Hence the total increment of energy in the whole system will be
As* — tArf — p a Ami — Mb Am\ — etc.
(516)
fA g Am) - )J h Amr h -~ etc. )
If the value of this expression is necessarily positive, for finite
changes as well as infinitesimal in the nature of the part of the film
to which Ae*, etc. relate,* the increment of energy of the whole
system will be positive for any possible changes in the nature of the
film, and the film will be stable, at least with respect to changes in
its nature, as distinguished from its position. For, if we write
De*, Drf, Dm* a , Dm*, etc., Dm*, Dmf, etc.
for the energy, etc. of any element of the surface of discontinuity, we
have from the supposition just made
ADe* - t A Drf -//„ A Dm* - /,,, ADmf - etc,
- fA g A Dm* - fA k ADm\ ~ etc. > ; (517)
and integrating for the whole surface, since
A/Dm.*=0, AfDml=0, etc.,
we have
A/Dt* - t A/Drf- pi a AfDml - ,u b A/Dm* - etc. > 0. (518)
Now A/Drf is the increment of the entropy of the whole surface,
and -A/Drf is therefore the increment of the entropy of the two
homogeneous masses. In like manner, —AfDml, —AfDml, etc.
are the increments of the quantities of the components in these masses.
The expression
_ t A/Drf - /J a A /Dm* - Mb A /Dm* - etc.
* In the case of infinitesimal changes in the nature of the film, the sign A must be
interpreted, as elsewhere in this paper, without neglect of infinitesimals of the higher
orders. Otherwise, by equation (501), the above expression would have the value
zero.
402 J. W. G-ibbs — Equilibrium of Heterogeneous Substances.
denotes therefore, according to equation (12), the increment of energy
of the two homogeneous masses, and since AfD& denotes the
increment of energy of the surface, the above condition expresses
that the increment of the total energy of the system is positive.
That we have only considered the possible formation of such films as
are capable of existing in equilibrium between the given homogeneous
masses can not invalidate the conclusion in regard to the stability of
the film, for in considering whether any state of the system will have
less energy than the given state, we need only consider the state of
least energy, which is necessarily one of equilibrium.
If the expression (516) is capable of a negative value for an infini-
tesimal change in the nature of the part of the film to which the
symbols relate, the film is obviously unstable.
If the expression is capable of a negative value, but only for finite
and not for infinitesimal changes in the nature of this part of the
film, the film is practically unstable* i. e., if such a change were
made in a small part of the film, the disturbance would tend to
increase. But it might be necessary that the initial disturbance
should also have a finite magnitude in respect to the extent of
surface in which it occurs ; for we cannot suppose that the thermo-
dynamic relations of an infinitesimal part of a surface of discontinuity
are independent of the adjacent parts. On the other hand, the
changes which we have been considering are such that every part
of the film remains in equilibrium with the homogeneous masses
on each side ; and if the energy of the system can be diminished by
a finite change satisfying this condition, it may perhaps be capable
of diminution by an infinitesimal change which does not satisfy the
same condition. We must therefore leave it undetermined whether
the film, which in this case is practically unstable, is or is not
unstable in the strict mathematical sense of the term.
Let us consider more particularly the condition of practical stabil-
ity, in which we need not distinguish between finite and infinitesimal
changes. To determine whether the expression (516) is capable of a
negative value, we need only consider the least value of which it is
capable. Let us write it in the fuller form
f s" _ £ s» _ t (y s " - if') - )A a (ml" - ml') - Mo (ml" — mf) - etc. )
- $ (mf - O - Ml (mf - mf) - etc., j (519)
where the single and double accents distinguish the quantities which
* With respect to the sense in which this term is used, compare page 133.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 403
relate to the first and second states of the film, the letters without
accents denoting those quantities which have the same value in both
states. The differential of this expression when the quantities distin
guished by double accents are alone considered variable, and the area
of the surface is constant, will reduce by (501) to the form
(jtf— /4) dm*" + (X' - K) dmf," + etc.
To make this incapable of a negative value, we must have
M" = K> u^ess mf'=0,
[a" = jj' k , unless m\" = 0.
In virtue of these relations and by equation (502), the expression
(519), i. e., (516), will reduce to
g" s — a' s,
which will be positive or negative according as
a" - a' (520)
is positive or negative.
That is, if the tension of the film is less than that of any other film
which can exist between the same homogeneous masses (which has
therefore the same values of t, ju a , pi b , etc.), and which moreover has
the same values of the potentials jj g , pi h , etc., so far as it contains the
substances to which these relate, then the first film will be stable.
But the film will be practically unstable, if any other such film has a
less tension. [Compare the expression (141), by which the practical
stability of homogeneous masses is tested.]
It is, however, evidently necessary for the stability of the surface
of discontinuity with respect to deformation, that the value of the
superficial tension should be positive. Moreover, since we have by
(502) for the surface of discontinuity
£ s — trf —)J a ml — pi b mf — etc. - jj g m s g — ju k mf - etc. = o' s,
and by (93) for the two homogeneous masses
s' — t rf + p v' — )J a mj — fx h m b ' — etc. = 0,
s" - trf + p v" —fi a m" — fj b mf — etc. = 0,
if we denote by
s, i], v, m a , m b , etc., m g , m h , etc.,
the total energy, etc. of a composite mass consisting of two such
homogeneous masses divided by such a surface of discontinuity, we
shall have by addition of these equations
404 j. W. Bibbs— Equilibrium of Heterogeneous Substances.
s - trj + pv — pi a m a - jA h m b - etc. — ju g m. g - jj h m h — etc. = a s.
Now if the value of a is negative, the value of the first member of
this equation will decrease as s increases, and may therefore be
decreased by making the mass to consist of thin alternate strata of
the two kinds of homogeneous masses which we are considering.
There will be no limit to the decrease which is thus possible with a
given value of v, so long as the equation is applicable, i. e., so long
as the strata have the properties of similar bodies in mass. But it
may easily be shown (as in a similar case on pages 131, 132) that
when the values of
h P, Ma, Mi, etc., p g , ju h , etc.
are regarded as fixed, being determined by the surface of discon-
tinuity in question, and the values of
s, //, m at m b , etc., m y , m,, , etc.
are variable and may be determined by any body having the given
volume v, the first member of this equation cannot have an infinite
negative value, and must therefore have a least possible value, which
will be negative, if any value is negative, that is, if a is negative.
The body determining e, 7/, etc. which will give this least value
to this expression will evidently be sensibly homogeneous. With
respect to the formation of such a body, the system consisting of the
two homogeneous masses and the surface of discontinuity with the
negative tension is by (53) (see also page 133) at least practically
unstable, if the surface of discontinuity is very large, so that it can
afford the requisite material without sensible alteration of the values
of the potentials. (This limitation disappears, if all the component
substances are found in the homogeneous masses.) Therefore in a
system satisfying the conditions of practical stability with respect to
the possible formation of all kinds of homogeneous masses, negative
tensions of the surfaces of discontinuity are necessarily excluded.
Let us now consider the condition which we obtain by applying
(516) to infinitesimal changes. The expression may be expanded as
before to the form (519), and then reduced by equation (502) to the
form
s( G »- G >) + m f (ju/ - Mg ') + mf ( M/ f - itf + etc.
That the value of this expression shall be positive when the quanti-
ties are determined by two films which differ infinitely little is a
necessary condition of the stability of the film to which the single
J. W. G-ibbs — Equilibrium of Heterogeneous Substances. 405
accents relate. But if one film is stable, the other will in general be
so too, and the distinction between the films with respect to stability
is of. importance only at the limits of stability. If all films for all
values of ju g , jj k , etc. are stable, or all within certain limits, it is evident
that the value of the expression must be positive when the quantities
are determined by any two infinitesimally different films within the
same limits. For such collective determinations of stability the
condition may be written
—sAg — m s g Afx g — m\ Ap k - etc. > 0,
or
^er<_ r g A Mg - r k A/j h - etc. (521)
On comparison of this formula with (508), it appears that within the
limits of stability the second and higher differential coefficients of the
tension considered as a function of the potentials for the substances
which are found only at the surface of discontinuity (the potentials
for the substances found in the homogeneous masses and the tempera-
ture being regarded as constant) satisfy the conditions which would
make the tension a maximum if the necessary conditions relative to
the first differential coefficients were fulfilled.
In the foregoing discussion of stability, the surface of discontinuity
is supposed plane. In this case, as the tension is supposed positive,
there can be no tendency to a change of form of the surface. We
now pass to the consideration of changes consisting in or connected
with motion and change of form of the surface of tension, which we
shall at first suppose to be and to remain spherical and uniform
throughout.
In order that the equilibrium of a spherical mass entirely sur-
rounded by an indefinitely large mass of different nature shall be
neutral with respect to changes in the value of r, the radius of the
sphere, it is evidently necessary that equation (500), which in this
may be written
2 =r(p>~p»), (522)
as well as the other conditions of equilibrium, shall continue to hold
true for varying values of r. Hence, for a state of equilibrium which
is on the limit between stability and instability, it is necessary that
the equation
2do'=(p' -p")dr + rdp'
shall be satisfied, when the relations between da, dp', and dr are
determined from the fundamental equations on the supposition that
-
406 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
the conditions of equilibrium relating to temperature and the poten-
tials remain satisfied. (The differential coefficients in the equations
which follow are to be determined on this supposition.) Moreover, if
i. e., if the pressure of the interior mass increases less rapidly (or
decreases more rapidly) with increasing radius than is necessary to
preserve neutral equilibrium, the equilibrium is stable. But if
dr dr
-^> 2 ^~P'+P"> ( 524 )
the equilibrium is unstable. In the remaining case, when
dp' do' , ,.
farther conditions are of course necessary to determine absolutely
whether the equilibrium is stable or unstable, but in general the
equilibrium will be stable in respect to change in one direction and
unstable in respect to change in the opposite direction, and is there-
fore to be considered unstable. In general, therefore, we may call
(523) the condition of stability.
When the interior mass and the surface of discontinuity are formed
entirely of substances which are components of the external mass, p'
and G cannot vary and condition (524) being satisfied the equili-
brium is unstable.
But if either the interior homogeneous mass or the surface of dis-
continuity contains substances which are not components of the
enveloping mass, the equilibrium may be stable. If there is but one
such substance, and we denote its densities and potential by y\, JP
and /i 1? the condition of stability (523) will reduce to the form
or, by (98) and (508),
(rV 1 '+^r i )^ < p"-p'. (526)
In these equations and in all which follow in the discussion of this
case, the temperature and the potentials /j 2 , yu 3 , etc. are to be
regarded as constant. But
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 407
which. represents the total quantity of the component specified by the
suffix, must be constant. It is evidently equal to
%nr*y x ' + 4 nr* F x .
Dividing by 4/T and differentiating, we obtain
( r s y x > + 2 r r t ) dr + i r 3 dy t ' + r 2 df \ = 0,
or, since y t ' and r t are functions of ju li
( r Yl > + 2 r,, * + g gi! + r gJ ^ = . (m)
By means of this equation, the condition of stability is brought to
the form
3 dji x dj2 1
If we eliminate r by equation (522), we have
p'—p" a
3 (^' -p") d}i x ~*~ 2crdjj 1
If p' and o~ are known in terms of t, jj. ± , yu 2 , etc., we may express the first
member of this condition in terms of the same variables and p". This
will enable us to determine, for any given state of the external mass,
the values of ja 1 which will make the equilibrium stable or unstable.
If the component to which y x ' and r x relate is found only at the
surface of discontinuity, the condition of stability reduces to
r t * dju ± ^ i
d6
J. + - U/U-. ■ j.
Since r. =z — _
d/ij 9
we may also write
r. dor ^ 1 e?log ff ^ I
-VdF<--2' or aJi^i\<-2- < 531 )
Again, if r i = and -^— l = 0, the condition of stability reduces to
3 y x ' 2 dji x .
y^^7> 1 - ( fi32 )
Since y , ' = -4— ,
we may also write
p>-p"d ri '^3> dlogy,' > 3' (533 )
Trans. Conn. Acad., Yol. III. 52 Nov., 187*7.
408 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
When r is large, this will be a close approximation for any values of
r x , unless y x ' is very small. The two special conditions (531) and
(533) might be derived from very elementary considerations.
Similar conditions of stability may be found when there are more
substances than one in the inner mass or the surface of discontin-
uity, which are not components of the enveloping mass. In this case,
we have instead of (526) a condition of the form
(r Vl ' + 2 r t ) p + (ry,'+2 F,) p + etc. <P " -p', (534)
trom which - — , - — , etc. may be eliminated by means of equations
derived from the conditions that
y 1 'v'+r i s, y 2 'v' + r 2 s, etc.
must be constant.
Nearly the same method may be applied to the following problem.
Two dffferent homogeneous fluids are separated by a diaphragm hav-
ing a circular orifice, their volumes being invariable except by the
motion of the surface of discontinuity, which adheres to the edge of
the orifice : — to determine the stability or instability of this surface
when in equilibrium.
The condition of stability derived from (522) may in this case be
written
d(p^- 2 /) cte dr
1 do' < dv' {P P) W (535)
where the quantities relating to the concave side of the surface of ten-
sion are distinguished by a single accent.
If both the masses are infinitely large, or if one which contains all
the components of the system is infinitely large, p'—p" and a will
be constant, and the condition reduces to
dr ^
The equilibrium will therefore be stable or unstable according as the
surface of tension is less or greater than a hemisphere.
To return to the general problem : — if we denote by x the part of
the axis of the circular orifice intercepted between the center of the
orifice and the surface of tension, by H the radius of the orifice, and
by V the value of v' when the surface of tension is plane, we shall
have the geometrical relations
R 2 = 2rx — x 2 ,
and v' = V + f it r 2 x — $ n R* (r - x)
= V + n r x 2 — ^ 7t X s .
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 409
By differentiation we obtain
(r — x) dx + x dr = 0,
and dv' = n x 2 dr -j- (2 7T r x — n x 2 ) dx ;
whence (r — x) dv' = — 7T r x % dr. (536)
By means of this relation, the condition of stability may be reduced
to the form
dp' dp" 2dff , , r — x
dv' ~W"r W < {P ~ P ] V&~&' ( 7)
Let us now suppose that the temperature and all the potentials ex-
cept one, /i,, are to be regarded as constant. This will be the case
when one of the homogeneous masses is very large and contains all
the components of the system except one, or when both these,
masses are very large and there is a single substance at the surface
of discontinuity which is not a component of either; also when
the whole system contains but a single component, and is exposed
to a constant temperature at its surface. Condition (537) will re-
duce by (98) and (508) to the form
(r/-r/ + ^)^<(y-/')^. (see)
But y,'V + ri "v" + r,s
(the total quantity of the component specified by the suffix) must be
constant ; therefore, since
2
dv" = — dv'. and ds = - dv'.
r
(«'lt + ' , 'fe !+a l : :)^ + (^'-^ + -? i > 8 - o - (639)
By this equation, the condition of stability is brought to the form
v' -/-i + v" ~p~ -f S -=-
djjy dju 1 dfx 1
When the substance specified by the suffix is a component of either
2 r dr
of the homogeneous masses, the terms and s -= — 1 may generally
/ Co jX ,
be neglected. When it is not a component of either, the terms y x ',
Yi'i v ' I * i v " j~ ma y °f course be cancelled, but we must not
apply the formula to cases in which the substance spreads over the
diaphragm separating the homogeneous masses.
410 J. W. Gibbs— Equilibrium of Heterogeneous Substances.
In the cases just discussed, the problem of the stability of certain
surfaces of tension has been solved by considering the case of neutral
equilibrium,— a condition of neutral equilibrium affording the equa-
tion of the limit of stability. This method probably leads as directly
as any to the result, when that consists in the determination of the
value of a certain quantity at the limit of stability, or of the relation
which exists at that limit between certain quantities specifying the
state of the system. But problems of a more general character may
require a more general treatment.
Let it be required to ascertain the stability or instability of a fluid
system in a given state of equilibrium with respect to motion of the
surfaces of tension and accompanying changes. It is supposed that
the conditions of internal stability for the separate homogeneous
masses are satisfied, as well as those conditions of stability for the
surfaces of discontinuity which relate to small portions of these
surfaces with the adjacent masses. (The conditions of stability
which are here supposed to be satisfied have been already discussed
in part and will be farther discussed hereafter.) The fundamental
equations for all the masses and surfaces occurring in the system are
supposed to be known. In applying the general criteria of stability
which are given on page 110, we encounter the following difficulty.
The question of the stability of the system is to be determined by
the consideration of states of the system which are slightly varied
from that of which the stability is in question. These varied states
of the system are not in general states of equilibrium, and the rela-
tions expressed by the fundamental equations may not hold true of
them. More than this, — if we attempt to describe a varied state of
the system by varied values of the quantities which describe the
initial state, if these varied values are such as are inconsistent with
equilibrium, they may fail to determine with precision any state of
the system. Thus, when the phases of two contiguous homogeneous
masses are specified, if these phases are such as satisfy all the condi-
tions of equilibrium, the nature of the surface of discontinuity (if with-
out additional components) is entirely determined ; but if the phases
do not satisfy all the conditions of equilibrium, the nature of the sur-
face of discontinuity is not only undetermined, but incapable of deter-
mination by specified values of such quantities as we have employed
to express the nature of surfaces of discontinuity in equilibrium. For
example, if the temperatures in contiguous homogeneous masses are
different, we cannot specify the thermal state of the surface of discon-
tinuity by assigning to it any particular temperature. It would be
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 411
necessary to give the law by which the temperature passes over from
one value to the other. And if this were given, we could make no
use of it in the determination of other quantities, unless the rate of
change of the temperature were so gradual, that at every point we.
could regard the thermodynamic state as unaffected by the change
of temperature in its vicinity. It is true that we are also ignorant in
respect to surfaces of discontinuity in equilibrium of the law of
change of those quantities which are different in the two phases in
contact, such as the densities of the components, but this, although
unknown to us, is entirely determined by the nature of the phases in
contact, so that no vagueness is occasioned in the definition of any of
the quantities which we have occasion to use with reference to such
surfaces of discontinuity.
It may be observed that we have established certain differential
equations, especially (497), in which only the initial state is neces-
sarily one of equilibrium. Such equations may be regarded as estab-
lishing certain properties of states bordering upon those of equilib-
rium. But these are properties which hold true only when we dis-
regard quantities proportional to the square of those which express
the degree of variation of the system from equilibrium. Such equa-
tions are therefore sufficient for the determination of the conditions of
equilibrium, but not sufficient for the determination of the conditions
of stability
We may, however, use the following method to decide the question
of stability in such a case as has been described.
Beside the real system of which the stability is in question, it will
be convenient to conceive of another system, to which we shall attri-
bute in its initial state the same homogeneous masses and surfaces of
discontinuity which belong to the real system. We shall also sup-
pose that the homogeneous masses and surfaces of discontinuity of
this system, which we may call the imaginary system, have the same
fundamental equations as those of the real system. But the imagin-
ary system is to differ from the real in that the variations of its state
are limited to such as do not violate the conditions of equilibrium
relating to temperature and the potentials, and that the fundamental
equations of the surfaces of discontinuity hold true for these varied
states, although the condition of equilibrium expressed by equation
(500) may not be satisfied.
Before proceeding farther, we must decide whether we are to
examine the question of stability under the condition of a constant
external temperature, or under the condition of no transmission of
412 J. W. Gibbs— Equilibrium of Heterogeneous Substances.
heat to or from external bodies, and in general, to what external
influences we are to regard the system as subject. It will be con-
venient to suppose that the exterior of the system is fixed, and that
neither matter nor heat can be transmitted through it. Other cases
may easily be reduced to this, or treated in a manner entirely
analogous.
Now if the real system in the given state is unstable, there must be
some slightly varied state in which the energy is less, but the entropy
and the quantities of the components the same as in the given state
and the exterior of the system unvaried. But it may easily be shown
that the given state of the system may be made stable by constrain-
ing the surfaces of discontinuity to pass through certain fixed lines
situated in the unvaried surfaces. Hence, if the surfaces of discon-
tinuity are constrained to pass through corresponding fixed lines in
the surfaces of discontinuity belonging to the varied state just men-
tioned, there must be a state of stable equilibrium for the system
thus constrained which will differ infinitely little from the given state
of the system, the stability of which is in question, and will have the
same entropy, quantities of components and exterior," but less energy.
The imaginary system will have a similar state, since the real and
imaginary systems do not differ in respect to those states which
satisfy all the conditions of equilibrium for each surface of discontin-
uity. That is, the imaginary system has a state, differing infinitely
little from the given state, and with the same entropy, quantities of
components, and exterior, but with less energy.
Conversely, if the imaginary system has such a state as that just
described, the real system will also have such a state. This may be
shown by fixing certain lines in the surfaces of discontinuity of the
imaginary system in its state of less energy and then making the
energy a minimum under the conditions. The state thus determined
will satisfy all the conditions of equilibrium for each surface of dis-
continuity, and the real system will therefore have a corresponding
state, in which the entropy, quantities of components, and exterior
will be the same as in the given state, but the energy less.
We may therefore determine whether the given system is or is not
unstable, by applying the general criterion of instability (V) to the
imaginary system.
If the system is not unstable, the equilibrium is either neutral or
stable. Of course we can determine which of these is the case by
reference to the imaginary system, since this determination depends
upon states of equilibrium, in regard to which the real and imaginary
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 413
systems do not differ. We may therefore determine whether the
equilibrium of the given system is stable, neutral, or unstable, by
applying the criteria (3)-(7) to the imaginary system.
The result which we have obtained may be expressed as follows : — ■
In applying to a fluid system which is in equilibrium, and of which
all the small parts taken separately are stable, the criteria of stable,
neutral, and unstable equilibrium, we may regard the system as
under constraint to satisfy the conditions of equilibrium relating to
temperature and the potentials, and as satisfying the relations ex-
pressed by the fundamental equations for masses and surfaces, even
when the condition of equilibrium relating to pressure [equation
(500)] is not satisfied.
It follows immediately from this principle, in connection with equa-
tions (501) and (86), that in a stable system each surface of tension
must be a surface of minimum area for constant values of the volumes
which it divides, when the other surfaces bounding these volumes
and the perimeter of the surface of tension are regarded as fixed ;
that in a system in neutral equilibrium each surface of tension will
have as small an area as it can receive by any slight variations under
the same limitations; and that in seeking the remaining conditions of
stable or neutral equilibrium, when these are satisfied, it is only
necessary to consider such varied surfaces of tension as have similar
properties with reference to the varied volumes and perimeters.
We may illustrate the method which has been described by apply-
ing it to a problem but slightly different from one already (pp. 408,
409) discussed by a different method. It is required to determine the
conditions of stability for a system in equilibrium, consisting of two
different homogeneous masses meeting at a surface of discontinuity,
the perimeter of which is invariable, as well as the exterior of the
whole system, which is also impermeable to heat.
To determine what is necessary for stability in addition to the
condition of minimum area for the surface of tension, we need only
consider those varied surfaces of tension which satisfy the same con-
dition. We may therefore regard the surface of tension as deter-
mined by v\ the volume of one of the homogeneous masses. But the
state of the system would evidently be completely determined by the
position of the surface of tension and the temperature and potentials,
if the entropy and the quantities of the components were variable ;
and therefore, since the entropy and the quantities of the components
are constant, the state of the system must be completely determined
by the position of the surface of tension. We may therefore regard
414 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
all the quantities relating to the system as functions of v', and the
condition of stability may be written
ds _ . ld 2 s ■ „
-=-, dv' + - —j- dv' 2 -4- etc, > 0,
dv 2 dv 2 ' ^ '
where a denotes the total energy of the system. Now the conditions
of equilibrium require that
ds
Hence, the general condition of stability is that
d 2 s ^
d&>°' ^
Now if we write a', a", e s for the energies of the two masses and of
the surface, we have by (86) and (501), since the total entropy and
the total quantities of the several components are constant,
da = da' + da" + da s — —p' dv' —p" dv" -f g ds,
or, since dv" = — dv',
de . ds . . s
---p> +p - +ff - (542)
Hence,
d 2 s dp' dp" da ds d 2 s
dv' 2 dv' dv' dv' dv' dv' 2 '' ^ '
and the condition of stability may be written
d 2 s dp' dp" do' ds
dV 2 > dv' " dV ~~ do'dv'- ^^
If we now simplify the problem by supposing, as in the similar case
on page 409, that we may disregard the variations of the tempera-
ture and of all the potentials except one, the condition will reduce to
d 2 s . / , ,. . 7 _, ds \ du.
The total quantity of the substance indicated by the suffix t is
ri 'v' + ri "v" + r lS .
Making this constant, we have
(r,' - r/ + r, * )^ + (^ + 4^ + .g)„ 1= o, M6)
The condition of equilibrium is thus reduced to the form
/ ' „ ' _i_ r ch V
djA 1 djA x ' d/ij
J. TK G-ibbs — Equilibrium of Heterogeneous Substances. 415
where '-=-. , and -=— , are to be determined from the form of the surface
av dv 9
of tension by purely geometrical considerations, and the other differ-
ential coefficients are to be determined from the fundamental equa-
tions of the homogeneous masses and the surface of discontinuity.
Condition (540) may be easily deduced from this as a particular case.
The condition of stability with reference to motion of surfaces of
discontinuity admits of a very simple expression when we can treat
the temperature and potentials as constant. This will be the case
when one or more of the homogeneous masses, containing together
all the component substances, may be considered as indefinitely large,
the surfaces of discontinuity being finite. For if we write 2 As for
the sum of the variations of the energies of the several homogeneous
masses, and 2As s for the sum of the variations of the energies of the
several surfaces of discontinuity, the condition of stability may be
written
2 As + ^zl£ s >0, (548)
the total entropy and the total quantities of the several components
being constant. The variations to be considered are infinitesimal,
but the character A signifies, as elsewhere in this paper, that the ex-
pression is to be interpreted without neglect of infinitesimals of the
higher orders. Since the temperature and potentials are sensibly con-
stant, the same will be true of the pressures and surface-tensions, and
by integration of (86) and (501) we may obtain for any homogeneous
mass
As — tArj — p Av + fx 1 Am 1 + fi 2 Am 2 -f etc.,
and for any surface of discontinuity
As s = t Arf + ff J 8 + }a\ Am\ + p* Am 2 -f etc.
These equations will hold true of finite differences, when t, p, a jx
yu„, etc. are constant, and will therefore hold true of infinitesimal dif-
ferences, under the same limitations, without neglect of the infinitesi-
mals of the higher orders. By substitution of these values, the condi-
tion of stability will reduce to the form
— 2{pAv) + 2(ffAs) > 0,
or 2(pAv) — 2(6 As) < 0. (549)
That is, the sum of the products of the volumes of the masses by
their pressures diminished by the sum of the products of the areas of
the surfaces of discontinuity by their tensions must be a maximum.
This is a purely geometrical condition, since the pressures and ten-
Trans. Conn. Aoad., Vol. III. 53 Nov., 1811.
416 J. W. Gribbs— Equilibrium of Heterogeneous /Substances.
sions are constant. This condition is of interest, because it is always
sufficient for stability with reference to motion of surfaces of discon-
tinuity. For any system may be reduced to the kind described by
putting certain parts of the system in communication (by means of
fine tubes if necessary) with large masses of the proper temperatures
and potentials. This may be done without introducing any new
movable surfaces of discontinuity. The condition (549) when
applied to the altered system is therefore the same as when applied
to the original system. But it is sufficient for the stability of the
altered system, and therefore sufficient for its stability if we diminish
its freedom by breaking the connection between the original system
and the additional parts, and therefore sufficient for the stability of
the original system.
On the Possibility of the Formation of a Fluid of different Phase
within any Homogeneous Fluid.
The study of surfaces of discontinuity throws considerable light
upon the subject of the stability of such homogeneous fluid masses
as have a less pressure than others formed of the same components
(or some of them) and having the s,ame temperature and the same
potentials for their actual components.*
In considering this subject, we must first of all inquire how far our
method of treating surfaces of discontinuity is applicable to cases in
which the radii of curvature of the surfaces are of insensible magni-
tude. That it sbould not be applied to such cases without limitation
is evident from the consideration that we have neglected the term
i(C 1 — C 2 )d(c l —c 2 ) in equation (494) on account of the magnitude
of the radii of curvature compared with the thickness of the non-
homogeneous film. (See page 390). When, however, only spherical
masses are considered, this term will always disappear, since C 1 and
C 2 will necessarily be equal.
Again, the surfaces of discontinuity have been regarded as separat-
ing homogeneous masses. But we may easily conceive that a globu-
lar mass (surrounded by a large homogeneous mass of different
nature) may be so small that no part of it will be homogeneous, and
that even at its center the matter cannot be regarded as having auy
phase of matter in mass. This, however, will cause no difficulty, if
we regard the phase of the interior mass as determined by the same
* See page 161, where the term stable is used (as indicated on page 159) in a less
strict sense than in the discussion which here follows.
J. W. Gibbs — Equilibrium of Heterogeneous /Substances. 417
relations to the exterior mass as in other cases. Beside the phase of
the exterior mass, there will always be another phase having the
same temperature and potentials, but of the general nature of the
small globule which is surrounded by that mass and in equilibrium
with it. This phase is completely determined by the system con-
sidered, and in general entirely stable and perfectly capable of realiza-
tion in mass, although not such that the exterior mass could exist in
contact with it at a plane surface. This is the phase which we are to
attribute to the mass which we conceive as existing within the divid-
ing surface.*
With this understanding with regard to the phase of the fictitious
interior mass, there will be no ambiguity in the meaning of any of
the symbols which we have employed, when applied to cases in
which the surface of discontinuity is spherical, however small the
radius may be. Nor will the demonstration of the general theorems
require any material modification. The dividing surface, which
determines the value of £ s , t/ s , m s 15 m|, etc., is as in other cases to be
placed so as to make the term i(C x + 6 7 1 )6 x (c 1 +c 2 ) in equation (494)
vanish, i. e., so as to make equation (497) valid. It has been shown
on pages 387-389 that when thus placed it will sensibly coincide
with the physical surface of discontinuity, when this consists of a
non-homogeneous film separating homogeneous masses, and having
radii of curvature which are large compared with its thickness. But
in regard to globular masses too small for this theorem to have any
application, it will be worth while to examine how far we may be
certain that the radius of the dividing surface will have a real and
positive value, since it is only then that our method will have any
natural application.
The value of the radius of the dividing surface, supposed spherical,
of any globule in equilibrium with a surrounding homogeneous
fluid may be most easily obtained by eliminating o' from equations
(500) and (502), which have been derived from (497), and contain
the radius implicitly. If we write r for this radius, equation (500)
may be written
2 = (p' -p").r, (550)
the single and double accents referring respectively to the interior
and exterior masses. If Ave write [e], [77], [m t ~\, [m 2 ], etc. for the
* For example, in applying our formulas to a microscopic globule of water in
steam, by the density or pressure of the interior mass we should understand, not the
actual density or pressure at the center of the globule, but the density of liquid water
(in large quantities) which has the temperature and potential of the steam.
418 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
excess of the total energy, entropy, etc. in and about the globular
mass above what would be in the same space if it were uniformly
filled with matter of the phase of the exterior mass, we shall have
necessarily with reference to the whole dividing surface
£ s = [e] _ „/ {8y > _ f/)? v& = M _ v , {W _ ? ^
m\ = [mj - v' (y, 1 - y/), m% = [m 2 ] - v' (y 2 ' - y 2 '% etc.,
where e v ', s v ", ?/ v ', 77/, y 1 ' i y 1 ', etc. denote, in accordance with our
usage elsewhere, the volume-densities of energy, of entropy, and of
the various components, in the two homogeneous masses. We may
thus obtain from equation (502)
ff s = [€]-, v' (V — ev") — t[ij] + tv' (7/y' - r/ v ")
~Mi[m 1 ]+Miv'(y 1 '-y 1 ' , )-/J 2 [m 2 ]+ M2 v'{y 2 '-y 2 ' , )-etc. (551)
But by (93),
p' ■— — Sy'+triv+P! rZ + ^^'+e^,
p"=- E Y "+tT ?v " + Ml ri " + M 2 r 2 / '+etc.
Let us also write for brevity
W= [f] — t [rf\ — yu t [mj — )A 2 [m 3 ] — etc. (552)
(It will be observed that the value of W is entirely determined by
the nature of the physical system considered, and that the notion of
the dividing surface does not in any way enter into its definition.)
We shall then have
6 s — TT.+ v' (p' —p"), (553)
or, substituting for s and v' their values in terms of r,
4 n r 2 6 = W + i n r 3 (p - p"), (554)
and eliminating 6 by (550),
*7rrz(p'-p") = W, (555)
/ 3 IF \*
r = \**&=7)l ' (556)
If we eliminate r instead (?, we have
16 7t 6 3 _,
3 (p' - P
3W(p'.—p')*\i
*='-- 16*
(558)
Now, if we first suppose the difference of the pressures in the homo-
geneous masses to be very small, so that the surface of discontin-
uity is nearly plane, since without any important loss of generality
J. W. Gibbs — Mjuilibrium of Heterogeneous Substances. 419
we may regard 6 as positive (for if 6 is not positive when p'=p", the
surface when plane would not be stable in regard to position, as
it certainly is, in every actual case, when the proper conditions are
fulfilled with respect to its perimeter), we see by (550) that the pres-
sure in the interior mass must be the greater; i. e., we may regard
Gt> p 1 —p\ and r as all positive. By (555), the value of W will
also be positive. But it is evident from equation (552), which defines
W, that the value of this quantity is necessarily real, in any possible
case of equilibrium, and can only become infinite when r becomes
infinite and p'=p". Hence, by (556) and (558), as p' —p" increases
from very small values, W, r, and a have single, real, and positive
values until they simultaneously reach the value zero. Within this
limit, our method is evidently applicable ; beyond this limit, if
such exist, it will hardly be profitable to seek to interpret the
equations. But it must be remembered that the vanishing of the
radius of the somewhat arbitrarily determined dividing surface may
not necessarily involve the vanishing of the physical heterogeneity.
It is evident, however, (see pp. 387-389,) that the globule must be-
come insensible in magnitude before r can vanish.
It may easily be shown that the quantity denoted by W is the
work which would be required to form (by a reversible process) the
heterogeneous globule in the interior of a very large mass having
initially the uniform phase of the exterior mass. For this work is
equal to the increment of energy of the system when the globule is
formed without change of the entropy or volume of the wdiole system
or of the quantities of the several components. Now \rf\, [m,], [^ 2 ],
etc. denote the increments of entropy and of the components in the
space where the globule is formed. Hence these quantities with the
negative sign will be equal to the increments of entropy and of the
components in the rest of the system. And hence, by equation (86),
- t [//] — }i x [m,] - jx 2 [m 2 ] - etc.
will denote the increment of energy in all the system except where
the globule is formed. But [s] denotes the increment of energy in
that part of the system. Therefore, by (552), W denotes the total
increment of energy in the circumstances supposed, or the work re-
quired for the formation of the globule.
The conclusions which may be drawn from these considerations
with respect to the stability of the homogeneous mass of the pres-
sure p" (supposed less than p\ the pressure belonging to a different
phase of the same temperature and potentials) are very obvious.
420 J. W. G-ibbs — Equilibrium of Heterogeneous Substances.
Within those limits within which the method used has been justified,
the mass in question must be regarded as in strictness stable with
respect to the growth of a globule of the kind considered, since W,
the work required for the formation of such a globule of a certain
size (viz., that which would be in equilibrium with the surrounding
mass), will always be positive. Nor can smaller globules be formed,
for they can neither be in equilibrium with the surrounding mass,
being too small, nor grow to the size of that to which W relates.
If, however, by any external agency such a globular mass (of the size
necessary for equilibrium) were formed, the equilibrium has already
(page 406) been shown to be unstable, and with the least excess in
size, the interior mass would tend to increase without limit except
that depending on the magnitude of the exterior mass. We may
therefore regard the quantity W as affording a kind of measure of
the stability of the phase to which p" relates. In equation (55*7) the
value of W is given in terms of a and p' -p". If the three funda-
mental equations which give (f,p', and p" in terms of the tempera-
ture and the potentials were known, we might regard the stability
( W) as known in terms of the same variables. It will be observed
that when p'=p" the value of W is infinite. If p' —p" increases
without greater changes of the phases than are necessary for such
increase, W will vary at first very nearly inversely as the square of
p'—p". If p'—p" continues to increase, it may perhaps occur that
IF reaches the value zero; but until this occurs the phase is certainly
stable with respect to the kind of change considered. Another kind
of change is conceivable, which initially is small in degree but may
be great in its extent in space. Stability in this respect or stability
in respect to continuous changes of phase has already been discussed
(see page 162), and its limits determined. These limits depend
entirely upon the fundamental equation of the homogeneous mass of
which the stability is in question. But with respect to the kind of
changes here considered, which are initially small in extent but great
in degree, it does not appear how we can fix the limits of stability
with the same precision. But it is safe to say that if there is such a
limit it must be at or beyond the limit at which o' vanishes. This
latter limit is determined entirely by the fundamental equation of the
surface of discontinuity between the phase of which the stability is
in question and that of which the possible formation is in question.
We have already seen that when o' vanishes, the radius of the divid-
ing surface and the work W vanish with it. If the fault in the
homogeneity of the mass vanishes at the same time, (it evidently
J. W. Gfibbs — Equilibrium of Heterogeneous Substances. 421
cannot vanish sooner,) the phase becomes unstable at this limit.
But if the fault in the homogeneity of the physical mass does not
vanish with r, 6 and W] — and no sufficient reason appears why this
should not be considered as the general case, — although the amount
of work necessary to upset the equilibrium of the phase is infinitesi-
mal, this is not enough to make the phase unstable. It appears
therefore that W is a somewhat one-sided measure of stability.
It must be remembered in this connection that the fundamental
equation of a surface of discontinuity can hardly be regarded as
capable of experimental determination, except for plane surfaces, (see
pp. 394, 395,) although the relation for spherical surfaces is in the
nature of things entirely determined, at least so far as the phases are
separately capable of existence. Yet the foregoing discussion yields
the following practical results. It has been shown that the real
stability of a phase extends in general beyond that limit (discussed
on pages 160, 161), which may be called the limit of practical stabil-
ity, at which the phase can exist in contact with another at a plane
surface, and a formula has been deduced to express the degree of
stability in such cases as measured by the amount of work necessary
to upset the equilibrium of the phase when supposed to extend indefi-
nitely in space. It has also been shown to be entirely consistent
with the principles established that this stability should have limits,
and the manner in which the general equations would accommodate
themselves to this case has been pointed out.
By equation (553), which may be written
W= 6 s - (p' ~ p") v', (559)
we see that the work W consists of two parts, of which one is always
positive, and is expressed by the product of the superficial tension
and the area of the surface of tension, and the other is always nega-
tive, and is numerically equal to the product of the difference of pres-
sure by the volume of the interior mass. We may regard the first
part as expressing the work spent in forming the surface of tension,
and the second part the work gained in forming the interior mass.*
* To make the physical significance of the above more clear, we may suppose the
two processes to be performed separately in the following manner. We may sup-
pose a large mass of the same phase as that which has the volume v' to exist
initially in the interior of the other. Of course, it must be surrounded by a resisting
envelop, on account of the difference of the pressures. We may, however, suppose
this envelop permeable to all the component substances, although not of such proper-
ties that a mass can form on the exterior like that within. We may allow the
422 J. W. G-ibbs — Equilibrium of Heterogeneous Substances.
Moreover, the second of these quantities, if we neglect its sign, is
always equal to two-thirds of the first, as appears from equation (550)
and the geometrical relation v'—%rs. We may therefore write
W= | as=i^{p' - p") v'. (500)
On the Possible Formation at the Surface where two different Homo-
geneous Fluids meet of a Fluid of different Phase from either.
Let A, B, and C be three different fluid phases of matter, which
satisfy all the conditions necessary for equilibrium when they meet
at plane surfaces. The components of A and B may be the same or
different, but C mast have no components except such as belong to A
or B. Let us suppose masses of the phases A and B to be separated
by a very thin sheet of the phase C. This sheet will not necessarily
be plane, but the sum of its principal curvatures must be zero. We
may treat such a system as consisting simply of masses of the phases
A and B with a certain surface of discontinuity, for in our previous
discussion there has been nothing to limit the thickness or the nature
of the film separating homogeneous masses, except that its thickness
has generally been supposed to be small in comparison with its radii
of curvature. The value of the superficial tension for such a film
will be C A c+ Gbc, if we denote by these symbols the tensions of the
surfaces of contact of the phases A and C, and B and C, respectively.
This not only appears from evident mechanical considerations, but
may also be easily verified by equations (502) and (93), the first of
which may be regarded as defining the quantity a. This value will
not be affected by diminishing the thickness of the film, until the
envelop to yield to the internal pressure until its contents are increased by v' without
materially affecting its superficial area. If this be done sufficiently slowly, the phase
of the mass within will remain constant. (See page 139.) A homogeneous mass of
the volume v' and of the desired phase has thus been produced, and the work gained
is evidently {p'—p")v'.
Let us suppose that a small aperture is now opened and closed in the envelop so as
to let out exactly the volume v of the mass within, the envelop being pressed inwards
in another place so as to diminish its contents by this amount. During the extrusion
of the drop and until the orifice is entirely closed, the surface of the drop must adhere
to the edge of the orifice, but not elsewhere to the outside surface of the envelop.
The work done in forming the surface of the drop will evidently be as or l{p'—p")v' .
Of this work, the amount (p'— p")v' will be expended in pressing the envelop inward,
and the rest in opening and closing the orifice. Both the opening and the closing
will be resisted by the capillary tension. If the orifice is circular, it must have, when
widest open, the radius determined by equation (550).
J. W. Gibbs — Equilibrium of Heterogeneous /Substances. 423
limit is reached at which the interior of the film ceases to have the
properties of matter in mass. Now if o~ AC + o~ BC is greater than C AB ,
the tension of the ordinary surface between A and B, such a film will
be at least practically unstable. (See page 403.) We cannot sup-
pose that o~ AB >o" AC + c BC , for this would make the ordinary surface
between A and B unstable and difficult to realize. If o" AB =rC AC + <7 BC ,
we may assume, in general, that this relation is not accidental, and
that the ordinary surface of contact for A and B is of the kind which
we have described.
Let us now suppose the phases A and B to vary, so as still to
satisfy the conditions of equilibrium at plane contact, but so that the
pressure of the phase C determined by the temperature and poten-
tials of A and B shall become less than the pressure of A and B. A
system consisting of the phases A and B will be entirely stable with
respect to the formation of any phase like C. (The case is not quite
identical with that considered on page 161, since the system in ques-
tion contains two different phases, but the principles involved are
entirely the same.)
With respect to variations of the phases A and B in the opposite
direction we must consider two cases separately. It will be conven-
ient to denote the pressures of the three phases by p A , p B , p c , and to
regard these quantities as functions of the temperature and potentials.
If o~ AB =o' AC -]-<T BC for values of the temperature and potentials which
make p A =p B =p c , it will not be possible to alter the temperature and
potentials at the surface of contact of the phases A and B so that
p A =p B , and Pc^>Pa, for the relation of the temperature and potentials
necessary for the equality of the three pressures will be preserved by
the increase of the mass of the phase C. Such variations of the phases
A and B might be brought about in separate masses, but if these
were brought into contact, there would be an immediate formation
of a mass of the phase C, with reduction of the phases of the adjacent
masses to such as satisfy the conditions of equilibrium with that
phase.
But if CTAB^o'Ac-f 0" BC , we can vary the temperature and potentials
so that^> A =^ B , and^> c ^>p A , and it will not be possible for a sheet of
the phase of C to form immediately, i. e., while the pressure of C is
sensibly equal to that of A and B ; for mechanical work equal to
^ac+^bc— o' AB per unit of surface might be obtained by bringing the
system into its original condition, and therefore produced without
any external expenditure, unless it be that of heat at the temperature
of the system, which is evidently incapable of producing the work.
Trans. Conn. Acad., Yol. III. 54 Nov., 1811.
424 J. W. Gibbs —Equilibrium of Heterogeneous Substances.
The stability of the system in respect to such a change must therefore
extend beyond the point where the pressure of C commences to be
Jess than that of A and B. We arrive at the same result if we use
the expression (520) as a test of stability. Since this expression has
a finite positive value when the pressures of the phases are all equal,
the ordinary surface of discontinuity must be stable, and it must
require a finite change in the circumstances of the case to make it
become unstable.*
In the preceding paragraph it is shown that the surface of contact
of phases A and B is stable under certain circumstances, with respect
to the formation of a thin sheet of the phase C. To complete the
demonstration of the stability of the surface with respect to the for-
mation of the phase C, it is necessary to show that this phase cannot
be formed at the surface in lentiform masses. This is the more neces-
sary, since it is in this manner, if at all, that the phase is likely to be
formed, for an incipient sheet of phase C would evidently be unstable
when o' AB <(7 AC + (f BC , and would immediately break up into lentiform
masses.
It will be convenient to consider first a lentiform mass of phase C
D in equilibrium between masses of phases A and B which
meet in a plane surface. Let figure 10 represent a section
of such a system through the centers of the spherical sur-
faces, the mass of phase A lying on the left ofDEH'FG,
and that of phase B on the right of DEH"F(1. Let
_|jj> the line joining the centers cut the spherical surfaces in
H' and H", and the plane of the surface of contact of A
and B in I. Let the radii of EH'F and EH"F be
denoted by r', r", and the segments I LI', I H" by x', x" .
Also let I E, the radius of the circle in which the spher-
ical surfaces intersect, he denoted by R. By a suitable
Fig. 3 0. application of the general condition of equilibrium we
may easily obtain the equation
r — x , r — x
^ac } ~r <^bc TJi — <5"ai
(561)
* It is true that such a ease as we are now considering is formally excluded in the
discussion referred to, which relates to a plane surface, and in which the system is
supposed thoroughly stable with respect to the possible formation of any different
homogeneous masses. Yet the reader will easily convince himself that the criterion
(520) is perfectly valid in this case with respect to the possible formation of a thin
sheet of the phase C, which, as we have seen, may be treated simply as a different
kind of surface of discontinuity.
J. IV. Gibbs — Equilibrium of Heterogeneous Substances. 425
which signifies that the components parallel to EF of the tension
ff AC and ff BC are together equal to <7 AB . If we denote by W the
amount of work which must be expended in order to form such a
lentiform mass as we are considering between masses of indefinite
extent having the phases A and B, we may write
W= M - JST, (562)
where M denotes the work expended in replacing the surface be-
tween A and B by the surfaces between A and C and B and C, and
AT denotes the work gained in replacing the masses of phases A and
B by the mass of phase C. Then
M— o- AC s AC -f ff BC s BC - <y AB s AB , (563)
where s AC , s BC , s AB denote the areas of the three surfaces concerned ;
and
JST= V (p c - p A ) + V" (pc ~Pb), (564)
where V and V" denote the volumes of the masses of the phases
A and B which are replaced. Now by (500),
2(5"ao -, 2<T R p
Pc —Pa = --— , and p c - p B — --p- (565) .
We have also the geometrical relations
V = f 7t r' 2 x' — i 7t B 2 (r' — x'),
V" = %7r r"* x" -\tz B 2 (r" - x").
By substitution we obtain
N— | 7i o- AC r' x' - f 7t B 2 - AC —,—
r
r " ry,"
+ I n o- BC r" x" -%7t B 2 cr BC — ~, (567)
and by (561),
JV= | 7t ff AC r'x' +±7t ff BC r" x" -%7tB* ff AB . (568)
Since
2 7t r 1 x = s AC , 2 tt r" x" = s BC , n B 2 — s AB ,
we may write
N= f (^ac s AC + o' BC s BC - G AB s AB ). (569)
(The reader will observe that the ratio of M and A r is the same as
that of the corresponding quantities in the case of the spherical mass
treated on pages 416-422.) We have therefore
"Wz=. i (ff AC s AC 4- a BC s BC - cr AB s AB ). (570)
This value is positive so long as
(566)
426 J. W. Gibbs — Equilibrium, of Heterogeneous Substances.
0" AC -+ O-bc > O'ab,
since s AC > s AB , and s BC > s AB .
But at the limit, when
0"ac + tf BC = (T AB ,
we see by (561) that
s ac — s AB , a ncl 5 BC = s AB ,
and therefore W = 0.
It should however be observed that in the immediate vicinity of the
circle in which the three surfaces of discontinuity intersect, the
physical state of each of these surfaces must be affected by the
vicinity of the others. We cannot, therefore, rely upon the formula
(570) except when the dimensions of the leiitiform mass are of sensi-
ble magnitude.
We may conclude that after we pass the limit at which p> c becomes
greater than p A and p B (supposed equal) lentiform masses of phase C
will not be formed until either o' AB =o' AC 4-o' BC , or p c -~p A becomes so
great that the lentiform mass which would be in equilibrium is one
of insensible magnitude. [The diminution of the radii with increas-
ing values of p c — p A is indicated by equation (565).] Hence, no
mass of phase C will be formed until one of these limits is reached.
Although the demonstration relates to a plane surface between A
and B, the result must be applicable whenever the radii of curvature
have a sensible magnitude, since the effect of such curvature may be
disregarded when the lentiform mass is of sufficiently small.
The equilibrium of the lentiform mass of phase C is easily proved
to be unstable, so that the quantity W affords a kind of measure of
the stability of plane surfaces of contact of the phases A and B.*
* If we represent phases by the position of points in such a manner that coexistent
phases (in the sense in which the term is used on page 152) are represented by the
same point, and allow ourselves, for brevity, to speak of the phases as having the
positions of the points by which they are represented, we may say that three coex-
istent phases are situated where three series of pairs of coexistent phases meet or
intersect. If the three phases are all fluid, or when the effects of solidity may be
disregarded, two cases are to be distinguished. Either the three series of coexistent
phases all intersect, — this is when each of the three surface-tensions is less than the
sum of the two others, — or one of the series terminates where the two others inter-
sect, — this is where one surface tension is equal to the sum of the others. The series
of coexistent phases will be represented by lines or surfaces, according as the phases
have one or two independently variable components. Similar relations exist when
the number of components is greater, except that they are not capable of geometrical
representation without some limitation, as that of constant temperature or pressure or
certain constant potentials.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 427
Essentially the same principles apply to the more general problem
in which the phases A and B have moderately different pressures, so
that their surfaces of contact must be curved, but the radii of curva-
ture have a sensible magnitude.
In order that a thin film of the phase C may be in equilibrium
between masses of the phases A and B, the following equations must
be satisfied —
<5"ac(c x +G 2 )=p A -p c ,
^BC^ +,0 2 ) = p C ~p B ,
where c x and c 2 denote the principal curvatures of the film, the
centers of positive curvature lying in the mass having the phase A.
Eliminating e 1 -{-c 2 , we have
0~bc (Pa ~ Pc) = 0"ac (Pc ~ Pb\
Pc =^±^. (M1)
°BCT °AC
It is evident that if p G has a value greater than that determined by
this equation, such a film will develop into a larger mass; if p c has a
less value, such a film will tend to diminish. Hence, when
the phases A and B have a stable surface of contact.
Again, if more than one kind of surface of discontinuity is possible
between A and B, for any given values of the temperature and poten-
tials, it will be impossible for that having the greater tension to dis-
place the other, at the temperature and with the potentials con-
sidered. Hence, when p c has the value determined by equation
(571), and consequently (y AC -\-o' BC is one value of the tension for the
surface between A and B, it is impossible that the ordinary tension
of the surface o~ AB should be greater than this. If <T AB =(r AC -{-.(T BC5
when equation (571) is satisfied, we may presume that a thin film of
the phase C actually exists at the surface between A and B, and that
a variation of the phases such as would make p c greater than the
second number of (571) cannot be brought about at that surface as
it would be prevented by the formation of a larger mass of the phase
C. But if cTabOac+cTbc when equation (571) is satisfied, this equa-
tion does not mark the limit of the stability of the surface between
A and B, for the temperature or potentials must receive a finite
"428 J.W. Gibbs— Equilibrium of Heterogeneous Substances.
change before the film of phase C, or (as we shall see in the following
paragraph) a lentiform mass of that phase, can be formed.
The work which mnst be expended in order to form on the surface
between indefinitely large masses of phases A and B a lentiform mass
of phase C in equilibrium, may evidently be represented by the
formula
" — °ac $ac + 0"bc $bc — Cab $ab
^ Pc V c + p A F A + p B V B , (573)
where £ AC , S BC denote the areas of the surfaces formed between A and
C, and B and C, S AB the diminution of the area of the surface between
A and B, V c the volume formed of the phase C, and V A , V B the
diminution of the volumes of the phases A and B. Let us now sup-
pose ff AC , C BC , C AB ,^ A ,^ B to remain constant and the external bound-
ary of the surface between A and B to remain fixed, while p c
increases and the surfaces of tension receive such alterations as are
necessary for equilibrium. It is not necessary that this should be
physically possible in the actual system ; we may suppose the changes
to take place, for the sake of argument, although involving changes
in the fundamental equations of the masses and surfaces considered.
Then, regarding W simply as an abbreviation for the second member
of the preceding equation, we have
dW—6 kC dS AC + o- BC dS BC — a AB dS AB
-p c d V c 4- p A d V A + p B d V B - V c dp c . (5 74)
But the conditions of equilibrium require that
^ac <ZS.Aa + C'bc ^£bc — Cab ^ab
—pcdV c -\-p A dV A +p B dV B =0. (575)
Hence,
dW— -Vcdpc. (576)
Now it is evident that V c will diminish as p c increases. Let us
integrate the last equation supposing p c to increase from its original
value until V c vanishes. This will give
W" — W = a negative quantity, (577)
where W and W" denote the initial and final values of W. But
W"=0. Hence W is positive. But this is the value of W in the
original system containing the lentiform mass, and expresses the
work necessary to form the mass between the phases A and B. It is
therefore impossible that such a mass should form on a surface be-
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 429
tween these phases. We must however observe the same limitation
as in the less general ease already discussed, — that; p c — p h , Pc~Pb
must not be so great that the dimensions of the lentiform mass are of
insensible magnitude. It may also be observed that the value of
these differences may be so small that there will not be room on the
surface between the masses of phases A and B for a mass of phase C
sufficiently large for equilibrium. In this case we may consider a
mass of phase C which is in equilibrium upon the surface between A
and B in virtue of a constraint applied to the line in which the three
surfaces of discontinuity intersect, which will not allow this line to
become longer, although not preventing it from becoming shorter.
We may prove that the value of W is positive by such an integra-
tion as we have used before.
Substitution of Pressures for Potentials in Fundamental Equations
for Surfaces.
The fundamental equation of a surface which gives the value of
the tension in terms of the temperature and potentials seems best
adapted to the purposes of theoretical discussion, especially when the
number of components is large or undetermined. But the experi-
mental determination of the fundamental equations, or the application
of any result indicated by theory to actual cases, will be facilitated
by the use of other quantities in place of the potentials, which shall
be capable of more direct measurement, and of which the numerical
expression (when the necessary measurements have been made) shall
depend upon less complex considerations. The numerical value of a
potential depends not only upon the system of units employed, but
also upon the arbitrary constants involved in the definition of the
energy and entropy of the substance to which the potential relates,
or, it may be, of the elementary substances of which that substance
is formed. (See page 152.) This fact and the want of means of
direct measurement may give a certain vagueness to the idea of the
potentials, and render the equations which involve them less fitted to
give a clear idea of physical relations.
Now the fundamental equation of each of the homogeneous masses
which are separated by any surface of discontinuity affords a relation
between the pressure in that mass and the temperature and potentials.
We are therefore able to eliminate one or two potentials from the
fundamental equation of a surface by introducing the pressures in
the adjacent masses. Again, when one of these masses is a gas-
430 J. W. G-ibbs— Equilibrium of Heterogeneous Substances.
mixture which satisfies Dalton's law as given on page 215, the
potential for each simple gas may be expressed in terms of the tem-
perature and the partial pressure belonging to that gas. By the
introduction of these partial pressures we may eliminate as many
potentials from the fundamental equation of the surface as there are
simple gases in the gas-mixture.
An equation obtained by such substitutions may be regarded as a
fundamental equation for the surface of discontinuity to which it
relates, for when the fundamental equations of the adjacent masses
are known, the equation in question is evidently equivalent to an
equation between the tension, temperature, and potentials, and we
must regard the knowledge of the properties of the adjacent masses
as an indispensable preliminary, or an essential part, of a complete
knowledge of any surface of discontinuity. It is evident, however,
that from these fundamental equations involving pressures instead
of potentials we cannot obtain by differentiation (without the use of
the fundamental equations of the homogeneous masses) precisely the
same relations as by the differentiation of the equations between the
tensions, temperatures, and potentials. It will be interesting to
inquire, at least in the more important cases, what relations may be
obtained by differentiation from the fundamental equations just
described alone.
If there is but one component, the fundamental equations of the
two homogeneous masses afford one relation more than is necessary
for the elimination of the potential. It may be convenient to regard
the tension as a function of the temperature and the difference of the
pressures. Now we have by (508) and (98)
da = — 7? s dt — r d).i j ,
d(j>'- P ") = (T/v'-O dt + (/-/) d Ml .
Hence we derive the equation
da = - (V s - --,-—„ (V - */)) dt _ ^-y, d (p' -p»), (578)
which indicates the differential coefficients of a with respect to t and
p' —p". For surfaces which may be regarded as nearly plane, it is
r
evident that —. T . represents the distance from the surface of ten-
r -r
sion to a dividing surface located so as to make the superficial
density of the single component vanish, (being positive, when the
J, Wl G-ibbs — Equilibrium of Heterogeneous Substances. 431
latter surface is on the side specified by the double accents,) and that
the coefficient of dt (without the negative sign) represents the super-
ficial density of entropy as determined by the latter dividing surface,
i. e., the quantity denoted by 7/ S(1) on page 397.
When there are two components, neither of which is confined to
the surface of discontinuity, we may regard the tension as a function
of the temperature and the pressures in the two homogeneous masses.
The values of the differential coefficients of the tension with respect
to these variables may be represented in a simple form if we choose
such substances for the components that in the particular state con-
sidered each mass shall consist of a single component. This will
always be possible when the composition of the two masses is not
identical, and will evidently not affect the values of the differential
coefficients. We then have
d0 = — r/ s dt — T\ dji l — F u dfx tl ,
dp' = 7] Y ' dt -f- y' d)A. l ,
dp" = 77/ dt -f y" dji u ,
where the marks ; and n are used instead of the usual , and s to indi-
cate the identity of the component specified with the substance of
the homogeneous masses specified by ' and " . Eliminating d/A j and
djj. u we obtain
dff = ~(y s -^, Vv '-^v Y ''\dt- 1 ^ / dp'- I ^d2/. (579)
We may generally neglect the difference ofp' and p", and write
(v* - 7V - p W') dt - (p + p) dp. (580)
The equation thus modified is strictly to be regarded as the equation
r r
for a plane surface. It is evident that —. and —. represent the dis-
y y
tances from the surface of tension of the two surfaces of which one
r r
would make F t vanish, and the other F in that — , -f- ~ represents
the distance between these two surfaces, or the diminution of vol-
ume due to a unit of the surface of discontinuity, and that the coeffi-
cient of dt (without the negative sign) represents the excess of
entropy in a system consisting of a unit of the surface of discon-
tinuity with a part of each of the adjacent masses above that
which the same matter would have if it existed in two homogeneous
masses of the same phases but without any surface of discontinuity.
Trans. Conn. Acad., Vol. III. 55 Nov., IS 1 ? 1 ;.
dff=-
Hi
432 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
(A mass thus existing without any surface of discontinuity must of
course be entirely surrounded by matter of the same phase.)*
The form in which the values of ( — ) and [—- ) are given in
\dtjp \dpjt s
equation (580) is adapted to give a clear idea of the relations of
these quantities to the particular state of the system for which they
are to be determined, but not to show how they vary with the state
of the system. For this purpose it will be convenient to have the
values of these differential coefficients expressed with reference to
ordinary components. Let these be specified as usual by t and 2 .
If we eliminate dfx x and d/J 2 from the equations
— do' — ?/ s dt -\- 1\ dju , -f- J \ dpt 2 ,
dp = 77 V ' dt + Yi dfi t + y 2 d/J 2 ,
dp = 7// dt -\-y ± " dji t + ?//, d/j 2 ,
* If we set
r. r
(a)
<P)
(<0
we may easily obtain, by means of equations (93) and (50?),
T£ a = tH s + (T—pV. (d)
Now equation (580) may be written
da = - H s dt + Vdp. (e)
Differentiating (d), and comparing the result with (e), we obtain
dE s - t d,R s — 2} dV. (/)
The quantities B s and H s might be called the superficial densities of energy and
entropy quite as properly as those which we denote by e s and t? s . In fact, when the
composition of both of the homogeneous masses is invariable, the quantities E s and
H are much more simple in their definition than e s and r/ s , and would probably be
more naturally suggested by the terms superficial density of energy and of entropy. It
would also be natural in this case to regard the quantities of the homogeneous masses
as determined by the total quantities of matter, and not by the surface of tension or
any other dividing surface. But such a nomenclature and method could not readily
be extended so as to treat cases of more than two components with entire generality.
In the treatment of surfaces of discontinuity in this paper, the definitions and
nomenclature which have been adopted will be strictly adhered to. The object of
this note is to suggest to the reader how a different method might be used in some
cases with advantage, and to show the precise relations between the quantities which
are used in this paper and others which might be confounded with them, and which
may be made more prominent when the subject is treated differently.
and in like manner
— — y~ y« .
H s
/S yf /V yll l\
r, r
K
yf y< T .
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 433
we obtain
I> C
d& = -dt + -j dp, (581)
where
A = y 1 "y 2 '-y 1 'y 2 % (582)
B= r /v > Vl ' y % ' , (583)
C=I\ (y/ - y 2 >) + r 2 ( ri > - Yl "). (584)
It will be observed that A vanishes when the composition of the two
homogeneous masses is identical, while JB and C do not, in general,
and. that the value of A is negative or positive according as the mass
specified by ' contains the component specified by x in a greater or
less proportion than the other mass. Hence, the values both of
l^r) and of (-—- ) become infinite when the difference in the com-
\dtjp \dpjt
position of the masses vanishes, and change sign when the greater-
proportion of a component passes from one mass to the other. This
might be inferred from the statements on page 155 respecting coex-
istent phases which are identical in composition, from which it ajypears
that when two coexistent phases have nearly the same composition,
a small variation of the temperature or pressure of the coexistent
phases will cause a relatively very great variation in the composition
of the phases. The same relations are indicated by the graphical
method represented in figure 6 on page 184.
With regard to gas-mixtures which conform to Dalton's law, we
shall only consider the fundamental equation for plane surfaces, and
shall suppose that there is not more than one component in the liquid
which does not appear in the gas-mixture. We have alreadv seen
that in limiting the fundamental equation to plane surfaces we can
get rid of one potential by choosing such a dividing surface that the
superficial density of one of the components vanishes. Let this be
done with respect to the component peculiar to the liquid, if such there
is ; if there is no such component, let it be done with respect to one
of the gaseous components. Let the remaining potentials be elim-
inated by means of the fundamental equations of the simple gases.
We may thus obtain an equation between the superficial tension, the
temperature, and the several pressures of the simple gases in the
gas-mixture or all but one of these pressures. Now, if we eliminate
d/u 2 , d/u 3 , etc. from the equations
-/ -/
434 J. W. Gibbs— Equilibrium of Heterogeneous /Substances.
dff = ~ ? 7s (1) dt - r 2(1) dpt z ^r 3(1) dM s yeic,
dp 2 = 77 V2 <$ -[- ^/ 2 dju 2 ,
dp 3 = 7v 3 ^ + y 3 dM s ,
etc.,
where the suffix x relates to the component of which the surface-
density has been made to vanish, and y 2 , y 37 etc denote the densities
of the gases specified in the gas mixture, and p 2 ,p s , etc., r/ V2 , r/ V3 ,
etc. the pressures and the densities of entropy due to these several
gases, we obtain
dff = - (? Ml) - ^ tj Y2 - ^ 7v3 - etc.) dt
_ EliD d p 2 - £»«> ^ 3 _ etc. (585)
This equation affords values of the differential coefficients of 6 with
respect to t,p 2 ,p s , etc., which may be set equal to those obtained
by differentiating the equation between these variables.
Thermal and Mechanical Relations pertaining to the Extension of a
Surface of Discontinuity.
The fundamental equation of a surface of discontinuity with one
or two component substances, beside its statical applications, is of
use to determine the heat absorbed when the surface is extended
under certain conditions.
Let us first consider the case in which there is only a single com-
ponent substance. We may treat the surface as plane, and place
the dividing surface so that the surface density of the single com-
ponent vanishes. (See page 397.) If we suppose the area of the
surface to be increased by unity without change of temperature or
of the quantities of liquid and vapor, the entropy of the whole will
be increased by t/ S(1) . Therefore, if we denote by Q the quantity of
heat which must be added to satisfy the conditions, we shall have
(? = *%a>, (586)
and by (514),
da da
It will be observed that the condition of constant quantities of
liquid and vapor as determined by the dividing surface which we
have adopted is equivalent to the condition that the total volume
shall remain constant.
J. W. Gibbs — Equilibrium, of Heterogeneous Substances. 435
Again, if the surface is extended without application of heat, while
the pressure in the liquid and vapor remains constant, the tempera-
ture will evidently be maintained constant by condensation of the
vapor. If we denote by M the mass of vapor condensed per unit of
surface formed, and by r/ u ' and r? u " the entropies of the liquid and
vapor per unit of mass, the condition of no addition of heat will
require that
M(? 7u "- Vnl ') = Vs{i) . (588)
The increase of the volume of liquid will be
%(i)
(589)
(590)
and the diminution of the volume of vapor
y" {Vm-VuY
Hence, for the work done (per unit of surface formed) by the exter-
nal bodies which maintain the pressure, we shall have
w =j»b^n}\ (591)
and, by (514) and (131),
dff dt d<5 d<j ,
W= —p--—= —p—=: - —- . 592)
dt dp dp dlogp '
The work expended directly in extending the film will of course be
equal to 6.
Let us now consider the case in which there are two component
substances, neither of which is confined to the surface. Since we can-
not make the superficial density of both these substances vanish by
any dividing surface, it will be best to regard the surface of tension
as the dividing surface. We may, however, simplify the formula by
choosing such substances for components that each homogeneous
mass shall consist of a single component. Quantities relating to
these components will be distinguished as on page 431. If the sur-
face is extended until its area is increased by unity, while heat is
added at the surface so as to keep the temperature constant, and the
pressure of the homogeneous masses is also kept constant, the phase
of these masses will necessarily remain unchanged, but the quantity
of one will be diminished by F /5 and that of the other by F u . Their
r r
entropies will therefore be diminished by — ' 7/ v ' and —§ r/ v ", respect-
436 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
ively. Hence, since the surface receives the increment of entropy tj^
the total quantity of entropy will be increased by
r r
Vs — -7 Vv ~ —g Vy ,
which by equation (580) is equal to
\dtjp'
Therefore, for the quantity of heat Q imparted to the surface, we
shall have
We must notice the difference between this formula and (587). In
(593) the quantity of heat Q is determined by the condition that the
temperature and pressures shall remain constant. In (587) these
conditions are equivalent and insufficient to determine the quantity
of heat. The additional condition by which Q is determined may be
most simply expressed by saying that the total volume must remain
constant. Again, the differential coefficient in (593) is defined by
considering p as constant; in the differential coefficient in (587) p
cannot be considered as constant, and no condition is necessary to
give the expression a definite value. Yet, notwithstanding the differ-
ence of the two cases, it is quite possible to give a single demonstra-
tion which shall be applicable to both. This may be done by con-
sidering a cycle of operations after the method employed by Sir
William Thomson, w T ho first pointed out these relations.*
The diminution of volume (per unit of surface formed) will be
and the work done (per unit of surface formed) by the external
bodies which maintain the pressure constant will be
. w = -*(*).= -(zzh); (595 >
Compare equation (592).
The values of Q and W may also be expressed in terms of quanti-
ties relating to the ordinary components. By substitution in (593)
and (595) of the values of the differential coefficients which are given
by (581), we obtain
* See Proc. Roy. Soc, vol. ix, p. 255, (June, 1858) ; or Phil. Mag., Ser. 4, vol. xvii,
p. 61.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 43 Y
0=-*|, W=-p~, (596)
where A, J?, and C represent the expressions indicated by (582)-
(584). It will be observed that the values of Q and IF are in general
infinite for the surface of discontinuity between coexistent phases
which differ infinitesimally in composition, and change sign with
the quantity A. When the phases are absolutely identical in
composition, it is not in general possible to counteract the effect of
extension of the surface of discontinuity by any supply of heat. For
the matter at the surface will not in general have the same composi-
tion as the homogeneous masses, and the matter required for the
increased surface cannot be obtained from these masses without
altering their phase. The infinite values of Q and W are explained
by the fact that when the phases are nearly identical in composition,
the extension of the surface of discontinuity is accompanied by the
vaporization or condensation of a very large mass, according as the
liquid or the vapor is the richer in that component which is necessary
for the formation of the surface of discontinuity.
If, instead of considering the amount of heat necessary to beep the
phases from altering while the surface of discontinuity is extended,
we consider the variation of temperature caused by the extension of
the surface while the pressures remain constant, it appears that this
variation of temperature changes sign with y x "y 2 ' — Y\y 2 " ■> ^ ut
vanishes with this quantity, i. e., vanishes when the composition of
the phases becomes the same. This may be inferred from the state-
ments on page 155, or from a consideration of the figure on page 184.
When the composition of the homogeneous masses is initially abso-
lutely identical, the effect on the temperature of a finite extension or
contraction of the surface of discontinuity will be the same, — either
of the two will lower or raise the temperature according as the tem-
perature is a maximum or minimum for constant pressure.
The effect of the extension of a surface of discontinuity which is
most easily verified by experiment is the effect upon the tension
before complete equilibrium has been reestablished throughout the
adjacent masses. A fresh surface between coexistent phases may be
regarded in this connection as an extreme case of a recently extended
surface. When sufficient time has elapsed after the extension of a
surface originally in equilibrium between coexistent phases, the
superficial tension will evidently have sensibly its original value,
unless there are substances at the surface which are either not found
438 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
at all in the adjacent masses, or are found only in quantities com-
parable to those in which they exist at the surface. But a surface
newly formed or extended may have a very different tension.
This will not be the case, however, when there is only a single
component substance, since all the processes necessary for equilibrium
are confined to a film of insensible thickness, and will require no
appreciable time for their completion.
When there are two components, neither of which is confined to the
surface of discontinuity, the reestablishment of equilibrium after the
extension of the surface does not necessitate any processes reaching
into the interior of the masses except the transmission of heat be-
tween the surface of discontinuity and the interior of the masses.
It appears from equation (593) that if the tension of the surface
diminishes with a rise of temperature, heat must be supplied to the
surface to maintain the temperature uniform when the surface is ex-
tended, i. e., the effect of extending the surface is to cool it; but if
the tension of any surface increases with the temperature, the effect
of extending the surface will be to raise its temperature. In either
case, it will be observed, the immediate effect of extending the sur-
face is to increase its tension. A contraction of the surface will of
course have the opposite effect. But the time necessary for the re-
establishment of sensible thermal equilibrium after extension or con-
traction of the surface must in most cases be very short.
In regard to the formation or extension of a surface between two
coexistent phases of more than two components, there are two ex-
treme cases which it is desirable to notice. When the superficial
density of each of the components is exceeding small compared with
its density in either of the homogeneous masses, the matter (as well
as the heat) necessary for the formation or extension of the normal
surface can be taken from the immediate vicinity of the surface with-
out sensibly changing the properties of the masses from which it is
taken. But if any one of these superficial densities has a considerable
value, while the density of the same component is very small in each
of the homogeneous masses, both absolutely and relatively to the
densities of the other components, the matter necessary for the for-
mation or extension of the normal surface must come from a consider-
able distance. Especially if we consider that a small difference of
density of such a component in one of the homogeneous masses will
probably make a considerable difference in the value of the corres-
ponding potential [see eq. (217)], and that a small difference in the
value of the potential will make a considerable difference in the ten-
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 439
sion [see eq. (508)], it will be evident that in this case a consider-
able time will be necessary after the formation of a fresh surface or
the extension of an old one for the reestablishment of the normal
value of the superficial tension. In intermediate cases, the reestab-
lishment of the normal tension will take pJace with different degrees
of rapidity.
But whatever the number of component substances, provided that
it is greater than one, and whether the reestablishment of equilibrium
is slow or rapid, extension of the surface will generally produce
increase and contraction decrease of the tension. It would evidently
be inconsistent with stability that the opposite effects should be pro-
duced. In general, therefore, a fresh surface between coexistent
phases has a greater tension than an old one.* By the use of fresh
surfaces, in experiments in capillarity, we may sometimes avoid the
effect of minute quantities of foreign substances, which may be
present without our knowledge or desire, in the fluids which meet at
the surface investigated.
When the establishment of equilibrium is rapid, the variation of
the tension from its normal value will be manifested especially during
the extension or contraction of the surface, the phenomenon resem-
bling that of viscosity, except that the variations of tension arising
from variations in the densities at and about the surface will be the
same in all directions, while the variations of tension due to any
property of the surface really analogous to viscosity would be great-
est in the direction of the most rapid extension.
We may here notice the different action of traces in the homogene-
ous masses of those substances which increase the tension and of
those which diminish it. When the volume-densities of a component
are very small, its surface-density may have a considerable positive
value, but can only have a very minute negative one.f For the
value when negative cannot exceed (numerically) the product of the
greater volume-density by the thickness of the non-homogeneous
* When, however, homogeneous masses which have not coexistent phases are
brought into contact, the superficial tension may increase with the course of time.
The superficial tension of a drop of alcohol and water placed in a large room will
increase as the potential for alcohol is equalized throughout the room, and is dimin-
ished in the vicinity of the surface of discontinuity.
f It is here supposed that we have chosen for components such substances as are
incapable of resolution into other components which are independently variable in the
homogeneous masses. In a mixture of alcohol and water, for example, the compo-
nents must be pure alcohol and pure water.
Trans. Conn. Acad., Vol. III. 56 Jan., 1878.
440 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
film. Each of these quantities is exceedingly small. The surface-
density when positive is of the same order of magnitude as the thick-
ness of the non-homogeneous film, but is not necessarily small com-
pared with other surface-densities because the volume-densities of
the same substance in the adjacent masses are small. Now the
potential of a substance which forms a very small part of a homo-
geneous mass certainly increases, and probably very rapidly, as the
proportion of that component is increased. [See (1 71) and (217).]
The pressure, temperature, and the other potentials, will not be
sensibly affected. [See (98).] But the effect on the tension of this
increase of the potential will be proportional to the surface-density,
and will be to diminish the tension when the surface-density is
positive. [See (508).] It is therefore quite possible that a very
small trace of a substance in the homogeneous masses should greatly
diminish the tension, but not possible that such a trace should greatly
increase it.*
Impermeable fflbns.
We have so far supposed, in treating of surfaces of discontinuity,
that they afford no obstacle to the passage of any of the component
substances from either of the homogeneous masses to the other. The
case, however, must be considered, in which there is a film of matter
at the surface of discontinuity which is impermeable to some or all of
* From the experiments of M. E. Duclaux, {Annates tie Ghimie et de Physique, Ser. 4,
vol. xxi, p. 383,) it appears that one per cent, of alcohol in water will diminish the
superficial tension to .933, the value for pure water being unity. The experiments do
not extend to pure alcohol, but the difference of the tensions for mixtures of alcohol
and water containing 10 and 20 per cent, water is comparatively small, the tensions
being .322 and .336 respectively.
According to the same authority (page 421 of the volume cited), one 3200th part of
Castile soap will reduce the superficial tension of water by one-fourth ; one 800th part
of soap by one-half. These determinations, as well as those relating to alcohol and
water, are made by the method of drops, the weight of the drops of different liquids
(from the same pipette) being regarded as proportional to their superficial tensions.
M. Athanase Dupre has determined the superficial tensions of solutions of soap by
different methods. A statical method gives for one part of common soap in 5000 of
water a superficial tension about one-half as great as for pure water, but if the tension
be measured on a jet close to the orifice, the value (for the same solution) is sensibly
identical with that of pure water. He explains these different values of the super-
ficial tension of the same solution as well as the great effect on the superficial tension
which a very small quantity of soap or other trifling impurity may produce, by the
tendency of the soap or other substance to form a film on the surface of the liquid.
(See Annales de, Chimie et de Physique, Ser. 4, vol. vii, p. 409, and vol. ix, p. 319.)
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 441
the components of the contiguous masses. Such may be the case,
for example, when a film of oil is spread on a surface of water, even
when the film is too thin to exhibit the properties of the oil in mass.
In such cases, if there is communication between the contiguous
masses through other parts of the system to which they belong, such
that the components in question can pass freely from one mass to the
other, the impossibility of a direct passage through the film may be
regarded as an immaterial circumstance, so far as states of equilib-
rium are concerned, and our formulae- will require no change. But
when there is no such indirect communication, the potential for any
component for which the film is impermeable may have entirely
different values on opposite sides of the film, and the case evidently
requires a modification of our usual method.
A single consideration will suggest the proper treatment of such
cases. If a certain component which is found on both sides of a film
cannot pass from either side to the other, the fact that the part of the
component which is on one side is the same kind of matter with the
part on the other side may be disregarded. All the general relations
must hold true, which would hold if they were really different sub-
stances. We may therefore write jj. t for the potential of the com-
ponent on one side of the film, and jj 2 for the potential of the same
substance (to be treated as if it were a different substance) on the
other side ; m\ for the excess of the quantity of the substance on the
first side of the film above the quantity which would be on that side
of the dividing surface (whether this is determined by the surface of
tension or otherwise) if the density of the substance were the same
near the dividing surface as at a distance, and m% for a similar quan-
tity relating to the other side of the film and dividing surface. On
the same principle, we may use F 1 and F 2 to denote the values of
m\ and m% per unit of surface, and m 1 ', m 2 \ y^, y 2 " to denote the
quantities of the substance and its densities in the two homogeneous
masses.
With such a notation, which may be extended to cases in which
the film is impermeable to any number of components, the equations
relating to the surface and the contiguous masses will evidently have
the same form as if the substances specified by the different suffixes
were all really different. The superficial tension will be a function
of }A t and yu 2 , with the temperature and the potentials for the other
components, and — 1\, —F 2 will be equal to its differential coeffi-
cients with respect to ju 1 and jj 2 . In a word, all the general rela-
tions which have been demonstrated may be applied to this case, if
442 J. W. G-ibbs— Equilibrium of Heterogeneous Substances.
we remember always to treat the component as a different substance
according as it is found on one side or the other of the impermeable
film.
When there is free passage for the component specified by the suf-
fixes 1 and 2 through other parts of the system, (or through any flaws
in the film,) we shall have in case of equilibrium ju l =pi 2 . If we wish
to obtain the fundamental equation for the surface when satisfying
this condition, without reference to other possible states of the sur-
face, we may set a single symbol for pi l and ju 2 in the more general
form of the fundamental equation. Cases may occur of an impermea-
bility which is not absolute, but which renders the transmission of
some of the components exceedingly slow. In such cases, it may be
necessary to distinguish at least two different fundamental equations,
one relating to a state of approximate equilibrium which may be
quickly established, and another relating to the ultimate state of
complete equilibrium. The former may be derived from the latter by
such substitutions as that just indicated.
The Conditions of Internal Equilibrium for a System of Hetero-
geneous Fluid Masses without neglect of the Influence of the
Surfaces of Discontinuity or of Gravity.
Let us now seek the complete value of the variation of the energy
of a system of heterogeneous fluid masses, in which the influence of
gravity and of the surfaces of discontinuity shall be included, and
deduce from it the conditions of internal equilibrium for such a sys-
tem. In accordance with the method which has been developed, the
intrinsic energy, (i. e., the part of the energy which is independent of
of gravity,) the entropy, and the quantities of the several compon-
ents must each be divided into two parts, one of which we regard as
belonging to the surfaces which divide approximately homogeneous
masses, and the other as belonging to these masses. The elements
of intrinsic energy, entropy, etc., relating to an element of surface
Ds will be denoted by De s , Drf, Dm\, Dm s 2 , etc., and those relating
to an element of volume Dv, by Dt v , Dif, Dm\, Dm\, etc. We
shall also use Dm* or rDs and Dm? or y Dv to denote the total
quantities of matter relating to the elements Ds and Dv respectively.
That is,
Dm s — TDs = Dm\ + Dm% -f etc., (597)
Dm? = y Dv = Dm\ -f Dm\ -f etc. (598)
The part of the energy which is due to gravity must also be divided
J. W. G-ibhs — Equilibrium of Heterogeneous Substances 443
into two parts, one of which relates to the elements Dm s , and the
other to the elements Dm v . The complete value of the variation of
the energy of the system will be represented by the expression
S/De v -f 6fD& -f dfg z Dm v -f- dfg z Dm s , (599)
in which g denotes the force of gravity, and z the height of the ele-
ment above a fixed horizontal plane.
It will be convenient to limit ourselves at first to the consideration
of reversible variations. This will exclude the formation of new
masses or surfaces. We may therefore regard any infinitesimal
variation in the state of the system as consisting of infinitesimal
variations of the quantities relating to its several elements, and
bring the sign of variation in the preceding formula after the sign
of integration. If w T e then substitute for dDs v , $Dt s , SDm v , SDm s ,
the values given by equations (13), (497), (597), (598), we shall have
for the condition of equilibrium with respect to reversible variations
of the internal state of the system
ft dJDrf -fp SDv +fMi SDm\ +ffl 2 8Dm\ + etc.
+ft SDrf -{-fff 6Ds+ffA t 6Dm\ +fju 2 6Dm% + etc.
+ fg 6z Dm y + fg z SDm\ +fgz dDm\ + etc.
-{-fg 6z Dm s + fgz 6 Dm* + fg z SDm% -f etc. = 0, (600)
Since equation (497) relates to surfaces of discontinuity which are
initially in equilibrium, it might seem that this condition, although
always necessary for equilibrium, may not always be sufficient. It is
evident, however, from the form of the condition, that it includes the
particular conditions of equilibrium relating to every possible deforma-
tion of the system, or reversible variation in the distribution of
entropy or of the several components. It therefore includes all the
relations between the different parts of the system which are neces-
sary for equilibrium, so far as reversible variations are concerned.
(The necessary relations between the various quantities relating to
each element of the masses and surfaces are expressed by the funda-
mental equation of the mass or surface concerned, or may be imme-
diately derived from it. See pp. 140-144 and 391-393.)
The variations in (600) are subject to the conditions which arise
from the nature of the system and from the supposition that the
changes in the system are not such as to affect external bodies. This
supposition is necessary, unless we are to consider the variations in
the state of the external bodies, and is evidently allowable in seeking
the conditions of equilibrium which relate to the interior of the sys-
444 J. W. Gibbs— Equilibrium of Heterogeneous Substances.
tem.* But before we consider the equations of condition in detail,
we may divide the condition of equilibrium (600) into the three condi-
tions
ftdDtf+ftdDrf=0, (601)
- fp SDv +/G dDs + fg Sz Dm v +fg Sz Dm s - 0, (602)
f;u 1 SDm\ +./>, SDm\ +fgzSDm\ + fgzSDm\
+ f/J 2 SDm Y 2 +fju 2 SDm%+fgzSDml^fgzSDm\
+ e.tc. = 0. (603)
For the variations which occur in any one of the three are evidently
independent of those which occur in the other two, and the equations
of condition will relate to one or another of these conditions sepa-
rately.
The variations in condition (601) are subject to the condition that
the entropy of the whole system shall remain constant. This may be
expressed by the equation
fSD? ? v +fSD?f = 0. (604)
To satisfy the condition thus limited it is necessary and sufficient that
t = const. (605)
throughout the whole system, which is the condition of thermal
equilibrium.
The conditions of mechanical equilibrium, or those that relate to
the possible deformation of the system, are contained in (602), which
may also be written
— fp SDv + / 6 SDs + fg y 6z Dv +fg F Sz Ds = 0. (606)
It will be observed that this condition has the same form as if the
different fluids were separated by heavy and elastic membranes with-
out rigidity and having at every point a tension uniform in all direc-
tions in the plane of the surface. The variations in this formula,
* "We have sometimes given a physical expression to a supposition of this kind, in
problems in which the peculiar condition of matter in the vicinity of surfaces of dis-
continuity was to be neglected, by regarding the system as surrounded by a rigid and
impermeable envelop. But the more exact treatment which we are now to give the
problem of equilibrium would require us to take account of the influence of the
envelop on the immediately adjacent matter. Since this involves the consideration of
surfaces of discontinuity between solids and fluids, and we wish to limit ourselves at
present to the consideration of the equilibrium of fluid masses, we shall give up the
conception of an impermeable envelop, and regard the system as bounded simply by a
imaginary surface, which is not a surface of discontinuity. The variations of the
system must be such as do not deform the surface, nor affect the matter external to it.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 445
beside their necessary geometrical relations, are subject to the condi-
tions that the external surface of the system, and the lines in which
the surfaces of discontinuity meet it, are fixed. The formula may be
reduced by any of the usual methods, so as to give the particular
conditions of mechanical equilibrium. Perhaps the following method
will lead as directly as any to the desired result.
It will be observed the quantities affected by d in (606) relate
exclusively to the position and size of the elements of volume and
surface into which the system is divided, and that the variations dp
and da do not enter into the formula either explicitly or implicitly.
The equations of condition which concern this formula also relate
exclusively to the variations of the system of geometrical elements,
and do not contain either dp or da. Hence, in determining whether
the first member of the formula has the value zero for every possible
variation of the system of geometrical elements, we may assign to
dp and da any values whatever, which may simplify the solution of
the problem, without inquiring whether such values are physically
possible.
Now when the system is in its initial state, the pressure^, in each
of the parts into which the system is divided by the surfaces of ten-
sion, is a function of the co-ordinates which determine the position of
the element Dv, to which the pressure relates. In the varied state
of the system, the element Dv will in general have a different position.
Let the variation dp be determined solely by the change in position
of the element Dv. This may be expressed by the equation
* = **-+**+** (6 M >
in which J: , 1. , -^ are determined by the function mentioned,
and dx, dy, dz by the variation of the position of the element Dv.
Again, in the initial state of the system the tension a, in each of
the different surfaces of discontinuity, is a function of two co-ordinates
go 1? cw s , which determine the position of the element Ds. In the
varied state of the system, this element will in general have a differ-
ent position. The change of position may be resolved into a com-
ponent lying in the surface and another normal to it. Let the varia-
tion da be determined solely by the first of these components of the
motion of Ds. This may be expressed by the equation
?^ da ,, , da <,
446 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
in which - — , _ — are determined by the function mentioned, and
aoo 1 d&) 2
$Ga u &gd 2 , by the component of the motion of Ds which lies in the
plane of the surface.
With this understanding, which is also to apply to dp and 6a
when contained implicitly in any expression, we shall proceed to the
reduction of the condition (606).
With respect to any one of the volumes into which the system is
divided by the surfaces of discontinuity, we may write
fp SDv = 6fp Dv -/dp Dv.
But it is evident that
dfpDu=fp6NDs,
where the second integral relates to the surfaces of discontinuity
bounding the volume considered, and oJV denotes the normal com-
ponent of the motion of an element of the surface, measured outward.
Hence,
fp 8Dv =fp SJVDs - fSp Dv.
Since this equation is true of each separate volume into which the
system is divided, we may write for the whole system
fp SDv =f{p'~p") oNDs - fdp Dv, (609)
where p' and p" denote the pressures on opposite sides of the element
Ds, and tfiVis measured toward the side specified by double accents.
Again, for each of the surfaces of discontinuity, taken separately,
f a 6Ds — dfo'Ds —fSff Ds,
and
SfffDs —fa (c, + c 2 ) SJVDs +/& 6TDI,
where c x and e 2 denote the principal curvatures of the surface,
(positive, when the centers are on the side opposite to that toward
which tfiVis measured,) Dl&n element of the perimeter of the surface,
and ST the component of the motion of this element which lies in the
plane of the surface and is perpendicular to the perimeter, (positive,
when it extends the surface). Hence we have for the whole system
fcrdDs=fc>{c l 4-c 2 ) SjYDs+f2((TdT)Dl-fd(TDs, (610)
where the integration of the elements Dl extends to all the lines in
which the surfaces of discontinuity meet, and the symbol ^ denotes
a summation with respect to the several surfaces which meet in such
a line.
By equations (609) and (610), the general condition of mechanical
equilibrium is reduced to the form
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 447
~f(p' ~P") SNDs -\-fSpDv +/6 (c, +c 2 ) SNDs
+f2{G ST) Dl —J'SciDs +fg y SzDv + fg FSzDs = 0.
Arranging and combining terms, we have
f{gy$z + Sp)Dv
_|_y [ {p n —p') diV-f o- (Cj+c.,) dN+gTdz ~ Sff]Ds
+ /2{ff6T)Di=0. (611)
To satisfy this condition, it is evidently necessary that the coefficients
of Dv, Ds, and Dl shall vanish throughout the system.
In order that the coefficient of Dv shall vanish, it is necessary and
sufficient that, in each of the masses into which the system is divided
by the surfaces of tension, p shall be a function of z alone, such that
±=-gy. (612)
In order that the coefficient of Ds shall vanish in all cases, it is
necessary and sufficient that it shall vanish for normal and for tan-
gential movements of the surface. For normal movements we may
write
So- = 0, and Sz = cos 5 dJV,
where 5 denotes the angle which the normal makes with a vertical
line. The first condition therefore gives the equation
p'— p"= '(G l -\-G 2 )-{- #r C 0s£, (613)
which must hold true at every point in every surface of discontinuity.
The condition with respect to tangential movements shows that in
each surface of tension 6 is a function of z alone, such that
*=,r. ( .H,
In order that the coefficient of Dl in (611) shall vanish, we
must have, for every point in every line in which surfaces of discon-
tinuity meet, and for any infinitesimal displacement of the line,
2(0ST)=O. (615)
This condition evidently expresses the same relations between the ten-
sions of the surfaces meeting in the line and the_ directions of per-
pendiculars to the line drawn in the planes of the various surfaces,
which hold for the magnitudes and directions of forces in equilibrium
in a plane.
In condition (603), the variations which relate to any component are
to be regarded as having the value zero in any part of the system in
Trans. Conn. Acad., Vol. III. 5Y Jan., 18^8.
448 J. W. Gibbs — Equilibrium, of Heterogeneous Substances.
which that substance is not an actual component.* The same is true
with respect to the equations of condition, which are of the form
f8JDml+fdDm\=Q, )
fdHmJ-\-fdI)ml = 0, I (616)
etc. )
(It is here supposed that the various components are independent, i. e.,
that none can be formed out of others, and that the parts of the sys-
tem in which any component actually occurs are not entirely sepa-
rated by parts in which it does not occur.) To satisfy the condition
(603), subject to these equations of condition, it is necessary and
sufficient that the conditions
^ 1 + gz = M 1 , j
M 2 +gz = M 2 , J- (617)
etc., J
(j¥j, M 2 , etc. denoting constants,) shall each hold true in those parts
of the system in which the substance specified is an actual component.
We may here add the condition of equilibrium relative to the possible
absorption of any substance (to be specified by the suffix a ) by parts
of the system of which it is not an actual component, viz., that the
expression fA a -\-gz must not have a less value in such parts of the
system than in a contiguous part in which the substance is an actual
component.
From equation (613) Math (605) and (61 7) we may easily obtain
the differential equation of a surface of tension (in the geometrical
sense of the term), when p\ p", and a are known in terms of the
temperature and potentials. For c 1 + c 2 and 3 may be expressed in
terms of the first and second differential coefficients of z with respect
to the horizontal co-ordinates, and p', p", o~, and F in terms of the
temperature and potentials. But the temperature is constant, and for
each of the potentials we may substitute — gz increased by a constant.
We thus obtain an equation in which the only variables are z and its
first and second differential coefficients with respect to the horizontal
co-ordinates. But it will rarely be necessary to use so exact a method.
Within moderate differences of level, we may regard y\ y", and 6 as
constant. We may then integrate the equation [derived from (612)]
d(p'-p") = g(y"~y r )dz,
* The term actual component has been defined for homogeneous masses on page 117,
and the definition may he extended to surfaces of discontinuity. It will he observed
that if a substance is an actual component of either of the masses separated by a sur-
face of discontinuity, it must be regarded as an actual component for that surface, as
well as when it occurs at the surface but not in either of the contiguous masses.
J. W. Gibhs — Equilibrium of Heterogeneous Substances. 449
which will give
p i~ p » = g(y»-y>)z, (618)
where z is to be measured from the horizontal plane for which p'=p".
Substituting this value in (613), and neglecting the term containing
F, we have
c i+ C z = ?Sr"'J^ z , (619)
6
where the coefficient of z is to be regarded as constant. Now the value
of z cannot be very large, in any surface of sensible dimensions, unless
y"-~y' is very small. We may therefore consider this equation as
practically exact, unless the densities of the contiguous masses are
very nearly equal. If we substitute for the sum of the curvatures
its value in terms of the differential coefficients of z with respect to
the horizontal rectangular co-ordinates, x and y, we have
/' dz 2 \ d' 2 z _ 2 dz dz d' 2 z /, efe z \ d' 2 z
\ dyijdx* ^dxdydxdy \ dx 2 )d^ g{y"-y')
dz 2 , dzM a Z - ib ^ Uj
\ dx 2 dy 2 )
With regard to the sign of the root in the denominator of the
fraction, it is to be observed that, if we always take the positive
value of the root, the value of the whole fraction will be positive or
negative according as the greater concavity is turned upward or
downward. But we wish the value of the fraction to be positive
when the greater concavity is turned toward the mass specified by a
single accent. We should therefore take the positive or negative
value of the root according as this mass is above or below the surface.
The particular conditions of equilibrium which are given in the
last paragraph but one may be regarded in general as the conditions
of chemical equilibrium between the different parts of the system,
since they relate to the separate components.* But such a designa-
tion is not entirely appropriate unless the number of components is
greater than one. In no case are the conditions of mechanical equi-
librium entirely independent of those which relate to temperature
and the potentials. For the conditions (612) and (614) may be re-
garded as consequences of (605) and (617) in virtue of the necessary
relations (98) and (508). f
* Concerning another kind of conditions of chemical equilibrium, which relate to
the molecular arrangement of the components, and not to their sensible distribution in
space, see pages 197-203.
f Compare page 206, where a similar problem is treated without regard to the influ-
ence of the surfaces of discontinuity.
450 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
The mechanical conditions of equilibrium, however, have an espe-
cial importance, since we may always regard them as satisfied in any
liquid (and not decidedly viscous) mass in which no sensible motions
are observable. In such a mass, when isolated, the attainment of
mechanical equilibrium will take place very soon; thermal and chem-
ical equilibrium will follow more slowly. The thermal equilibrium
will generally require less time for its approximate attainment than
the chemical ; but the processes by which the latter is produced will
generally cause certain inequalities of temperature until a state of
complete equilibrium is reached.
When a surface of discontinuity has more components than one
which do not occur in the contiguous masses, the adjustment of the
potentials for these components in accordance with equations (617)
may take place very slowly, or not at all, for want of sufficient
mobility in the components of the surface. But when this surface
has only one component which does not occur in the contiguous
masses, and the temperature and potentials in these masses satisfy
the conditions of equilibrium, the potential for the component pecu-
liar to the surface will very quickly conform to the law expressed in
(617), since this is a necessary consequence of the condition of
mechanical equilibrium (614) in connection with the conditions
relating to temperature and the potentials which we have supposed
to be satisfied. The necessary distribution of the substance peculiar
to the surface will be brought about by expansions and contractions
of the surface. If the surface meets a third mass containing this
component and no other which is foreign to the masses divided bv
the surface, the potential for this component in the surface will of
course be determined by that in the mass which it meets.
The particular conditions of mechanical equilibrium (612)-(615),
which may be regarded as expressing the relations which must sub-
sist between contiguous portions of a fluid system in a state of
mechanical equilibrium, are serviceable in determining whether a
given system is or is not in such a state. But the mechanical theo-
rems which relate to finite parts of the system, although they may
be deduced from these conditions by integration, may generally be
more easily obtained by a suitable application of the general condi-
tion of mechanical equilibrium (606), or by the application of ordi-
nary mechanical principles to the system regarded as subject to the
forces indicated by this equation.
It will be observed that the conditions of equilibrium relating to
temperature and the potentials are not affected by the surfaces of
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 451
discontinuity. [Compare (228) and (234).] * Since a phase cannot
vary continuously without variations of the temperature or the
potentials, it follows from these conditions that the phase at any
point in a fluid system which has the same independently variable
components throughout, and is in equilibrium under the influence of
gravity, must be one of a certain number of phases which are com-
pletely determined by the phase at any given point and the difference
of level of the two points considered. If the phases throughout the
fluid system satisfy the general condition of practical stability for
phases existing in large masses (viz., that the pressure shall be the least
consistent with the temperature and potentials), they will be entirely
determined by the phase at any given point and the differences of
level. (Compare page 210, where the subject is treated without
regard to the influence of the surfaces of discontinuity.)
Conditions of equilibrium relating to irreversible changes. — The
conditions of equilibrium relating to the absorption by any part of
the system of substances which are not actual components of that part
have been given on page 448. Those relating to the formation of new
masses and surfaces are included in the conditions of stability relat-
ing to such changes, and are not always distinguishable from them.
They are evidently independent of the action of gravity. We have
already discussed the conditions of stability with respect to the for-
mation of new fluid masses within a homogeneous fluid and at the
surface when two such masses meet (see pages 416-429), as well as
the condition relating to the possibility of a change in the nature of
a surface of discontinuity. (See pages 400-403, where the surface
considered is plane, but the result may easily be extended to curved
surfaces.) We shall hereafter consider, in some of the more import-
ant cases, the conditions of stability with respect to the formation of
new masses and surfaces which are peculiar to lines in which several
surfaces of discontinuity meet, and points in which several such lines
meet.
Conditions of stability relating to the whole system. — Beside the
conditions of stability relating to very small parts of a system, which
are substantially independent of the action of gravity, and are dis-
cussed elsewhere, there are other conditions, which relate to the
* If the fluid system is divided into separate masses by solid diaphragms which are
permeable to all the components of the fluids independently, the conditions of equi-
librium of the fluids relating to temperature and the potentials will not be affected.
(Compare page 139.) The propositions which follow in the above paragraph may be
extended to this case.
452 J. W. Gibbs— Equilibrium of Heterogeneous Substances.
whole system or to considerable parts of it. To determine the ques-
tion of the stability of a given fluid system under the influence of
gravity, when all the conditions of equilibrium are satisfied as well
as those conditions of stability which relate to small parts of the sys-
tem taken separately, we may use the method described on page
413, the demonstration of which (pages 411, 412) will not require
any essential modification on account of gravity.
When the variations of temperature and of the quantities M t , M s ,
etc. [see (617)] involved in the changes considered are so small that
they may be neglected, the condition of stability takes a very simple
form, as we have already seen to be the case with respect to a sys-
tem uninfluenced by gravity. (See page 415.)
We have to consider a varied state of the system in which the
total entropy and the total quantities of the various components are
unchanged, and all variations vanish at the exterior of the system,—
in which, moreover, the conditions of equilibrium relating to tem-
perature and the potentials are satisfied, and the relations expressed
by the fundamental equations of the masses and surfaces are to be
regarded as satisfied, although the state of the system is not one of
complete equilibrium. Let us imagine the state of the system to vary
continuously in the course of time in accordance with these condi-
tions and use the symbol d to denote the simultaneous changes which
take place at any instant. If we denote the total energy of the
system by E, the value of dE may be expanded like that of 6E in
(599) and (600), and then reduced (since the values of t, ju^+gz,
M 2 -\-gz, etc. are uniform throughout the system, and the total entropy
and total quantities of the several components are constant) to the
form
dE — -fp dDv +fg dz J)m v +fo' dDs +fg dz Dm s
= -fp dDv +fg y dz Dv +fff dDs +fg F dz Ds, (621)
where the integrations relate to the elements expressed by the symbol
D. The value of p at any point in any of the various masses, and
that of o' at any point in any of the various surfaces of discontinuity
are entirely determined by the temperature and potentials at the
point considered. If the variations of t and M lf M 2 , etc. are to be
neglected, the variations of p and 6 will be determined solely by the
change in position of the point considered. Therefore, by (612) and
(614),
dp=—gydz, daz= g rdz;
and
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 453
dE= —fp dDv -fdp Dv +fc> dDs +fdcf Ds
= - dfp Dv + dfff Ds. (622)
If we now integrate with respect to d, commencing at the given
state of the system, we obtain
AE=- AfpDv + AfeDs, (623)
where A denotes the value of a quantity in a varied state of the sys-
tem diminished by its value in the given state. This is true for finite
variations, and is therefore true for infinitesimal variations without
neglect of the infinitesimals of the higher orders. The condition of
stability is therefore that
AfpDv-Af(?Ds<Q, (624)
or that the quantity
fpDv—fffDs (625)
has a maximum value, the values of p and &, for each different mass
or surface, being regarded as determined functions of z. (In ordin-
ary cases 6 may be regarded as constant in each surface of discon-
tinuity, and j>asa linear function of z in each different mass.) It
may easily be shown (compare page 416) that this condition is always
sufficient for stability with reference to motion of surfaces of discon-
tinuity, even when the variations of t, M 1 , 3f 2 , etc. cannot be neg-
lected in the determination of the necessary condition of stability
with respect to such changes.
On the Possibility of the Formation of a New Surface of Discon-
tinuity where several Surfaces of Discontinuity meet.
When more than three surfaces of discontinuity between homo-
geneous masses meet along a line, we may conceive of a new surface
being formed between any two of the masses which do not meet in a
surface in the original state of the system. The condition of stability
with respect to the formation of such a surface may be easily obtained
by the consideration of the limit between stability and instability, as
exemplified by a system which is in equilibrium when a very small
surface of the kind is formed.
To fix our ideas, let us suppose that there are four homogeneous
masses A, B, C, and D, which meet one another in four surfaces,
which we may call A-B, B-C, C-D, and D-A, these surfaces all meeting
along a line L. This is indicated in figure 11 by a section of the sur-
454 J. W. Gibbs— Equilibrium of Heterogeneous Substances.
faces cutting the line L at right angles at a point O. In an infini-
tesimal variation of the state of the system, we may conceive of a
small surface being formed betweeu A and C (to be called A-C),
so that the section of the surfaces of discontinuity by the same
plane takes the form indicated in figure 12. Let us suppose that
Pig. 11.
the condition of equilibrium (615) is satisfied both for the line L in
which the surfaces of discontinuity meet in the original state of the
system, and for the two such lines (which we may call L' and L") in the
varied state of the system, at least at the points O' and O" where they
are cut by the plane of section. We may therefore form a quadri-
lateral of which the sides a/3, /3y, yd, 6a are equal in numerical
value to the tensions of the several surfaces A-B, B-C, CD, D-A,
and are parallel to the normals to these surfaces at the point O
in the original state of the system. In like manner, for the varied
state of the system we can construct two triangles having similar
relations to the surfaces of discontinuity meeting at O' and O".
But the directions of the normals to the surfaces A-B and B-C
at O' and to C-D and D-A at 0" in the varied state of the system
differ infinitely little from the directions of the corresponding nor-
mals at O in the initial state. We may therefore regard a/3, {3y
as two sides of the triangle representing the surfaces meeting at O',
and y6, 6a as two sides of the triangle representing the surfaces
meeting at O". Therefore, if we join ay, this line will represent the
direction of the normal to the surface A-C, and the value of its ten-
sion. If the tension of a surface between such masses as A and C had
been greater than that represented by ay, it is evident that the initial
state of the system of surfaces (represented in figure 1 1 ) would have
been stable with respect to the possible formation of any such sur-
face. If the tension had been less, the state of the system would
have been at least practically unstable. To determine whether it is
unstable in the strict sense of the term, or whether or not it is prop-
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 455
erly to be regarded as in equilibrium, would require a more refined
analysis than we have used.*
The result which we have obtained may be generalized as follows.
When more than three surfaces of discontinuity in a fluid system meet
in equilibrium along a line, with respect to the surfaces and masses
immediately adjacent to any point of this line we may form a polygon
of which the angular points shall correspond in order to the different
masses separated by the surfaces of discontinuity, and the sides to
these surfaces, each side being perpendicular to the corresponding
surface, and equal to its tension. With respect to the formation of
new surfaces of discontinuity in the vicinity of the point especially
considered, the system is stable, if every diagonal of the polygon is
less, and practically unstable, if any diagonal is greater, than the
tension which would belong to the surface of discontinuity between
the corresponding masses. In the limiting case, when the diagonal
is exactly equal to the tension of the corresponding surface, the sys-
tem may often be determined to be unstable by the application of
the principle enunciated to an adjacent point of the line in which the
surfaces of discontinuity meet. But when, in the polygons con-
structed for all points of the line, no diagonal is in any case greater
* We may here remark that a nearer approximation in the theory of equilibrium and
stability might be attained, by taking special account, in our general equations, of the
lines in which surfaces of discontinuity meet. These lines might be treated in a
manner entirely analogous to that in which we have treated surfaces of discontinuity.
"We might recognize linear densities of energy, of entropy, and of the several sub-
stances which occur about the line, also a certain linear tension. With respect to
these quantities and the temperature and potentials, relations would hold analogous to
those which have been demonstrated for surfaces of discontinuity. (See pp. 393-393.)
If the sum of the tensions of the lines 1/ and L", mentioned above, is greater than the
tension of the line L, this line mil be in strictness stable (although practically unstable)
with respect to the formation of a surface between A and C, when the tension of such
a surface is a little less than that represented by the diagonal ay.
The different use of the term practically unstable in different parts of this paper need
not create confusion, since the general meaning of the term is in all cases the same.
A system is called practically unstable when a very small (not necessarily indefinitely
small) disturbance or variation in its condition will produce a considerable change.
In the former part of this paper, in which the influence of surfaces of discontinuity
was neglected, a system was regarded as practically unstable when such a result
would be produced by a disturbance of the same order of magnitude as the quantities
relating to surfaces of discontinuity which were neglected. But where surfaces of
discontinuity are considered, a system is not regarded as practically unstable, unless
the disturbance which will produce such a result is very small compared with the
quantities relating to surfaces of discontinuity of any appreciable magnitude.
Trans. Conn. Acad., Vol. III. 58 March, 1878.
456 J. W. G-ibbs — Equilibrium of Heterogeneous Substances.
than the tension of the corresponding surface, but a certain diagonal
is equal to the tension in the polygons constructed for a finite portion
of the line, farther investigations are necessary to determine the
stability of the system. For this purpose, the method described on
page 413 is evidently applicable.
A similar proposition may be enunciated in many cases with re-
spect to a point about which the angular space is- divided into solid
angles by surfaces of discontinuity. If these surfaces are in equilib-
rium, we can always form a closed solid figure without re-entrant
angles of which the angular points shall correspond to the several
masses, the edges to the surfaces of discontinuity, and the sides to
the lines in which these edges meet, the edges being perpendicular
to the corresponding surfaces, and equal to their tensions, and the
sides being perpendicular to the corresponding lines. Now if the
solid angles in the physical system are such as may be subtended by
the sides and bases of a triangular prism enclosing the vertical point,
or can be derived from such by deformation, the figure representing
the tensions will have the form of two triangular pyramids on oppo-
site sides of the same base, and the system will be stable or practic-
ally unstable with respect to the formation of a surface between the
masses which only meet in a point, according as the tension of a sur-
face between such masses is greater or less than the diagonal joining
the corresponding angular points of the solid representing the ten-
sions. This will easily appear on consideration of the case in which
a very small surface between the masses would be in equilibrium.
The Conditions of Stability for Fluids relating to the Formation
of a New Phase at a Fine in which Three Surfaces of
Discontinuity meet.
With regard to the formation of new phases there are particular
conditions of stability which relate to lines in which several surfaces
of discontinuity meet. We may limit ourselves to the case in which
there are three such surfaces, this being the only one of frequent occur-
rence, and may treat them as meeting in a straight line. It will be
convenient to commence by considering the equilibrium of a system
in which such a line is replaced by a filament of a different phase.
Let us suppose that three homogeneous fluid masses, A, B, and C,
are separated by cylindrical (or plane) surfaces, A-B, B-C, C-A, which
at first meet in a straight line, each of the surface-tensions c AB , c BC , g ck
being less than the sum of the other two. Let us suppose that the
J. W. G-ibbs — Equilibrium of Heterogeneous Substances. 457
system is then modified by the introduction of a fourth fluid mass D,
which is placed between A, B, and C, and is separated from them by
cylindrical surfaces D-A, D-B, D-C meeting A-B, B-C, and C-A in
straight lines. The general form of the surfaces is shown by figure
14, in which the full lines represent a section perpendicular to all the
surfaces. The system thus modified is to be in equilibrium, as well
as the original system, the position of the surfaces A-B, B-C, C-A
being unchanged. That the last condition is consistent with equili-
brium will appear from the following mechanical considerations.
Fig. 14. Fig. 15. Fig. 16.
Let Vv denote the volume of the mass D per Unit of length or the area
of the curvilinear triangle a be. Equilibrium is evidently possible for
any values of the surface-tensions (if only ff AB , ff BC , <j ca satisfy the con-
dition mentioned above, and the tensions of the three surfaces meet-
ing at each of the edges of D satisfy a similar condition) with any
value (not too large) of v D , if the edges of D are constrained to remain
in the original surfaces A-B, B-C, and C-A, or these surfaces extended,
if necessary, without change of curvature. (In certain cases one of
the surfaces D-A, D-B, D-C may disappear and D will be bounded
by only two cylindrical surfaces.) We may therefore regard the
system as maintained in equilibrium by forces applied to the edges
of D and acting at right angles to A-B, B-C, C-A. The same forces
would keep the system in equilibrium if D were rigid. They must
therefore have a zero resultant, since the nature of the mass D is im-
material when it is rigid, and no forces external to the system would
be required to keep a corresponding part of the original system in
equilibrium. But it is evident from the points of application and
directions of these forces that they cannot have a zero resultant unless
each force is zero. We may therefore introduce a fourth mass D
without disturbing the parts which remain of the surfaces A-B, B-C,
C-D.
It will be observed that all the angles at a, b, c, and d in figure 14
are entirely determined by the six surface-tensions C AB , G" BC , g'ca? °da,
#db, g"dc- [See (615).] The angles maybe derived from the tensions
V
458 J.W. G-ibbs — Equilibrium of Heterogeneous Substances.
by the following construction, which will also indicate some important
properties. If we form a triangle a (3 y (figure 15 or 16) having sides
equal to o~ AB , <r BC , a CM the angles of the triangle will be supplements
of the angles at d. To fix our ideas, we may suppose the sides of the
triangle to be perpendicular to the surfaces at d. Upon (3 y we may
then construct (as in figure 16) a triangle (3 y 8' having sides equal
to ff BCi <r DC} <r DB , upon y a a triangle y a 8" having sides equal to
°ca, ^da, <5"dc, and upon a j3 a triangle a/36'" having sides equal to
0ab> ^db, C DA . These triangles are to be on the same sides of the lines
§ Y, Y a, at. f3, respectively, as the triangle a [3 y. The angles of
these triangles will be supplements of the angles of the surfaces of
discontinuity at a, b, and c. Thus (3 y 6'=dab, and a y 6"=dba.
Now if 6' and 8" fall together in a single point 8 within the triangle
<x/3y, d"' will fall in the same point, as in figure 15. In this
case we shall have j3 y 6 -f a y 8=a y j3, and the three angles of the
curvilinear triangle adb will be together equal to two right angles.
The same will be true of the three angles of each of the triangles
bdc, cda, and hence of the three angles of the triangle a be. But
if 8', 8", 8'" do not fall together in the same point within the triangle
a (3 y, it is either possible to bring these points to coincide within
the triangle by increasing some or all of the tensions cr DA , o- DB , 0"dc,
or to effect the same result by diminishing some or all of these ten-
sions. (This will easily appear when one of the points 8', 8", 8'" falls
within the triangle, if we let the two tensions which determine this
point remain constant, and the third tension vary. When all the
points 8\ 8", 6'" fall without the triangle a /3 y, we may suppose the
greatest of the tensions o" DA , <r DB , <7 DC — the two greatest, when these
are equal, and all three when they all are equal— to diminish until
one of the points 8', 6", 8'" is brought within the triangle a /3 y.)
In the first case we may say that the tensions of the new surfaces are
too small to be represented by the distances of an internal point from
the vertices of the triangle representing the tensions of the original
surfaces (or, for brevity, that they are too small to be represented as
in figure 15) ; in the second case we may say that they are too great
to be thus represented. In the first case, the sum of the angles in
each of the triangles adb, bdc, cda is less than two right angles
(compare figures 14 and 16): in the second case, each pair of the
triangles a fi 8'", fly 6", y a 8" will overlap, at least when the ten-
sions <T DA , or DB , G^c are only a little too great to be represented as in
figure 15, and the sum of the angles of each of the triangles ad b,
bd c, c da will be greater than two right angles.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 459
Let us denote by v A , v B , v c the portions of v D which were originally
occupied by the masses A, B, C, respectively, by s DA , s BB , s DC , the
areas of the surfaces specified per unit of length of the mass D, and
by s AB , *bcj s ca, the areas of the surfaces specified which were replaced
by the mass D per unit of its length. In numerical value, v A , v B , v c
will be equal to the areas of the curvilinear triangles bed, cad,
abd; and s DA , s DB , s vc , s AB , s BC , s CA to the lengths of the lines be, ca,
ab, cd, ad, b d. Also let
W s = 0' DA S BA + C DB S m -+- 0' DC S DC — <J AB s AB — 0~ BC «bc — Gca s c A , (626)
and TPv=^ D y D — p A v A - p B v B — p c v c . (62V)
The general condition of mechanical equilibrium for a system of
homogeneous masses not influenced by gravity, when the exterior of
the whole system is fixed, may be written
2 {<? ds) - 2 (p dv) = 0. (628)
[See (606).] If we apply this both to the original system consisting
of the masses A, B, and C, and to the system modified by the intro-
duction of the mass D, and take the difference of the results, suppos-
ing the deformation of the system to be the same in each case, we
shall have
0DA # «DA "I- <5"dB # *DB + <5"dC <^BC ~ ^AB # *AB — ^BC #«BC
— tfcA #*CA - Pd 3vv + p A <5v A + p B 6v B + p c dv c = 0. (629)
In view of this relation, if we differentiate (626) and (62V) regarding
all quantities except the pressures as variable, we obtain
dW&—dW V = S DA dGvA + «DB <^DB + «DC ^DC
- «ab da AB - s BC d6 BC — s CA dff CA . (630)
Let us now suppose the system to vary in size, remaining always
similar to itself in form, and that the tensions diminish in the same
ratio as lines, while the pressures remain constant. Such changes
will evidently not impair the equilibrium. Since all the quantities
«da, <Tda, *db, ^db, etc. vary in the same ratio,
s^ A do- VA =^d{0^ A s^ A ), s DB ^DB = i^(o'DB«DB), etc. (631)
We have therefore by integration of (630)
W s - W v = i(0'T) A S I)A J r (T- DB S X)B + 0' DC S DC —0' AB S AB -0' BC S BC -0' CA S CA ), (632)
whence, by (626),
W s = 2 W y , (633)
The condition of stability for the system when the pressures and
tensions are regarded as constant, and the position of the surfaces
460 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
A-B, B-C, C-A as fixed, is that W s — W Y shall be a minimum under
the same conditions. [See (549).] Now for any constant values of
the tensions and of p A ,p B , p c , we may make v n so small that when it
varies, the system remaining in equilibrium, (which will in general
require a variation of p D ,) we may neglect the curvatures of the lines
da, db, d c, and regard the figure abed as remaining similar to
itself. For the total curvature (i. e., the curvature measured in
degrees) of each of the lines a b, be, ca may be regarded as con-
stant, being equal to the constant difference of the sum of the angles
of one of the curvilinear triangles adb, b d c, c da and two right
angles. Therefore, when v D is very small, and the system is so
deformed that equilibrium would be preserved if p^ had the proper
variation, but this pressure as well as the others and all the tensions
remain constant, W s will vary as the lines in the figure abed, and
Wv as the square of these lines. Therefore, for such deformations,
W v oc W s *.
This shows that the system cannot be stable for constant pressures
and tensions when v D is small and W v is positive, since W s — W v
will not be a minimum. It also shows that the system is stable
when W v is negative. For, to determine whether W s — W v is a
minimum for constant values of the pressures and tensions, it will
evidently be sufficient to consider such varied forms of the system as
give the least value to W s — W Y for any value of u D in connection
with the constant pressures and tensions. And it may easily be
shown that such forms of the system are those which would pre-
serve equilibrium if p D had the proper value.
These results will enable us to determine the most important ques-
tions relating to the stability of a line along which three homogene-
ous fluids A, B, C meet, with respect to the formation of a different
fluid D. The components of D must of course be such as are found
in the surrounding bodies. We shall regard p B and (fj) A , cr DB , o' DC as
determined by that phase of D which satisfies the conditions of equi-
librium with the other bodies relating to temperature and the
potentials. These quantities are therefore determinable, by means
of the fundamental equations of the mass D and of the surfaces D-A,
D-B, D-0, from the temperature and potentials of the given system.
Let us first consider the case in which the tensions, thus deter-
mined, can be represented as in figure 15, and jt? D has a value con-
sistent with the equilibrium of a small mass such as we have been
considering. It appears from the preceding discussion that when v B is
J. W. O-ibbs — Equilibrium of Heterogeneous Substances. 461
sufficiently small the figure abed may be regarded as rectilinear, and
that its angles are entirely determined by its tensions. Hence the
ratios of v A , -y B , v c , v-o> for sufficiently small values of* v D , are deter-
mined by the tensions alone, and for convenience in calculating these
ratios, we may suppose p A , p B , p c to be equal, which will make the
figure abed absolutely rectilinear, and make p D equal to the other
pressures, since it is supposed that this quantity has the value neces-
sary for equilibrium. We may obtain a simple expression for the
ratios of v A , v B , v c , ^d in terms of the tensions in the following
manner. We shall write [D B C], [D C A], etc., to denote the areas
of triangles having sides equal to the tensions of the surfaces between
the masses specified.
v A : v B :: triangle b d c : triangle a d c
: : be sin bed : ac sin acd
: : sin bac sin bed : sin abe sin acd
: : sin ySfi sin dafi : sin yd a sin d/3a
: : sin yd [3 d/3 : sin yd a da
: : triangle y d /3: triangle y d a
: : [D B C] : [D C A].
Hence,
« a :» b :d c :« d ::[DBC]:[DCA]:[DAB]:[ABC], (634)
where
W[{&AB+(?BC+cr CA )(c> AB +(} B c—0' CA )(C> BO +C> CA --C> AB ) {ff CA +ff AB - 0' BC )]
may be written for [A B C], and analogous expressions for the other
symbols, the sign */ denoting the positive root of the necessarily posi-
tive expression which follows. This proportion will hold true in any
case of equilibrium, when the tensions satisfy the condition mentioned
and Vj) is sufficiently small. Now if p A =.p B =p c , p^ will have the
same value, and we shall have by (627) W Y = 0, and by (633) W s = 0.
But when v D is very small, the value of W s is entirely determined by
the tensions and v D . Therefore, whenever the tensions satisfy the
condition supposed, and v D is very small (whether p A , p Bi p c are
equal or unequal,)
0= W s =W v =p D v D — p A v A -p B v B -p c v c , (635)
which with (634) gives
_ [D B C]p A + [D C A]^ B + [D A B] Pc
lD ~ [DBC] + [DCA] + [DAB] ' l ;
Since this is the only value of$> D for which equilibrium is possible when
462 J. W. G-ibbs — Equilibrium of Heterogeneous Substances.
the tensions satisfy the condition supposed and v D is small, it follows
that when p D has a less value, the line where the fluids A, B, C meet
is stable with respect to the formation of the fluid D. When/> D has
a greater value, if such a line can exist at all, it must be at least
practically unstable, i. e., if only a very small mass of the fluid D
should be formed it would tend to increase.
Let us next consider the case in which the tensions of the
new surfaces are too small to be represented as in figure 15. If
the pressures and tensions are consistent with equilibrium for any
very small value of -y D , the angles of each of the curvilinear tri-
angles adb, bdc, c da will be together less than two right angles,
and the lines a b, bo, oa, will be convex toward the mass D. For
given values of the pressures and tensions, it will be easy to deter-
mine the magnitude of -y D . For the tensions will give the total
curvatures (in degrees) of the lines ab, be, ca; and the pressures
will give the radii of curvature. These lines are thus completely
determined. In order that ?j d shall be very small it is evidently
necessary that p D shall be less than the other pressures. Yet if the
tensions of the new surfaces are only a very little too small to be
represented as in figure 15, u D may be quite small when the value
of p D is only a little less than that given by equation (636). In any
case, when the tensions of the new surfaces are too small to be repre-
sented as in figure 15, and w D is small, W v is negative, and the equi-
librium of the mass D is stable. Moreover, W s — W v , which repre-
sents the work necessary to form the mass D with its surfaces in
place of the other masses and surfaces, is negative.
With respect to the stability of a line in which the surfaces A-B,
B-C C-A meet, when the tensions of the new surfaces are too small to
be represented as in figure 15, we first observe that when the pressures
and tensions are such as to make v D moderately small but not so
small as to be neglected, [this will be when p D is somewhat smaller
than the second member of (636), — more or less smaller according as
the tensions differ more or less from such as are represented in
figure 15,] the equilibrium of such a line as that supposed (if it is
capable of existing at all) is at least practically unstable. For greater
values of p D (with the same values of the other pressures and the
tensions) the same will be true. For somewhat smaller values of p D ,
the mass of the phase D which will be formed will be so small, that
we may neglect this mass and regard the surfaces A-B, B-C, C-A as
meeting in a line in stable equilibrium. For still smaller values of
p D , we may likewise regard the surfaces A-B, B-C, C-A as capable
J. W. G-ibbs — Equilibrium of Heterogeneous Substances. 463
of meeting in stable equilibrium. It may be observed that when
w D , as determined by our equations, becomes quite insensible,
the conception of a small mass D having the properties deducible
from our equations ceases to be accurate, since the matter in the
vicinity of a line where these surfaces of discontinuity meet must
be in a peculiar state of equilibrium not recognized by our equations.*
But this cannot affect the validity of our conclusion with respect to
the stability of the line in question.
The case remains to be considered in which the tensions of the new
surfaces are too great to be represented as in figure 15. Let us sup-
pose that they are not very much too great to be thus represented.
When the pressures are such as to make v D moderately small (in case
of equilibrium) but not so small that the mass D to which it relates
ceases to have the properties of matter in mass, [this will be when
Pu is somewhat greater than the second member of (636), — more or
less greater according as the tensions differ more or less from such as
are represented in figure 15,] the line where the surfaces A-B, B-C,
C-A meet will be in stable equilibrium with respect to the formation
of such a mass as we have considered, since Ws— Wy will be posi-
tive. The same will be true for less values of p B . For greater values
of $> D , the value of W s - W y , which measures the stability with respect
to the kind of change considered, diminishes. It does not vanish,
according to our equations, for finite values of jt? D . But these equa-
tions are not to be trusted beyond the limit at which the mass D
ceases to be of sensible magnitude.
But when the tensions are such as we now suppose, we must also
consider the possible formation of a mass D within a closed figure in
which the surfaces D-A, D-B, D-C meet together (with the surfaces
A-B, B-C, C-A) in two opposite points. If such a figure is to be in
equilibrium, the six tensions must be such as can be represented by
* See note on page 455. We may here add that the linear tension there mentioned
may have a negative value. This would be the case with respect to a line in which
three surfaces of discontinuity are regarded as meeting, but where nevertheless there
really exists in stable equilibrium a filament of different phase from the three sur-
rounding masses. The value of the linear tension for the supposed line, would be
nearly equal to the value of W & — W y for the actually existing filament. (For the
exact value of the linear tension it would be necessary to add the sum of the linear
tensions of the three edges of the filament.) We may regard two soap-bubbles
adhering together as an example of this case. The reader will easily convince himself
that in an exact treatment of the equilibrium of such a double bubble we must recog-
nize a certain negative tension in the line of intersection of the three surfaces of
discontinuity.
Trans. Conn. Acad., Vol. III. 59 March, 18?8.
464 J. W. Gibbs — Equilibrium of Heterogeneous /Substances.
the six distances of four points in space (see page 455), — a condition
which evidently agrees with the supposition which we have made. If
we denote by w Y the work gained in forming the mass D (of such size
and form as to be in equilibrium) in place of the other masses, and by
w s the work expended in forming the new surfaces in place of the old,
it may easily be shown by a method similar to that used on page 459
that w s =z^w v . From this we obtain w s — v) v =^w v . This is evidently
positive when p D is greater than the other pressures. But it diminishes
with increase of p B , as easily appears from the equivalent expression
%w s . Hence the Hne of intersection of the surfaces of discontinuity A-B ,
B-C, C-A is stable for values of p B greater than the other pressures
(and therefore for all values of p^) so long as our method is to be re-
garded as accurate, which will be so long as the mass D which would
be in equilibrium has a sensible size.
In certain cases in which the tensions of the new surfaces are much
too large to be represented as in figure 15, the reasoning of the two
last paragraphs will cease to be applicable. These are cases in which
the six tensions cannot be represented by the sides of a tetrahedron.
It is not necessary to discuss these cases, which are distinguished by
the different shape which the mass D would take if it should be
formed, since it is evident that they can constitute no exception to
the results which we have obtained. For an increase of the values of
Cda 5 ^db, ^dc cannot favor the formation of D, and hence cannot im-
pair the stability of the line considered, as deduced from our equa-
tions. Nor can an increase of these tensions essentially affect the
fact that the stability thus demonstrated may fail to be realized when
pj) is considerably greater than the other pressures, since the a priori
demonstration of the stability of any one of the surfaces A-B, B-C,
C-A, taken singly, is subject to the limitation mentioned. (See page
426.)
The Condition of Stability for Fluids relating to the Formation
of a JYew Phase at a Point where the Vertices of
Four Different Masses meet.
Let four different fluid masses A, B, C, D meet about a point, so as
to form the six surfaces of discontinuity A-B, B-C, C-A, DA, D-B,
D-C, which meet in the four lines A-B-C, B-C-D, C-D-A, D-A-B, these
lines meeting in the vertical point. Let us suppose the system stable
in other respects, and consider the conditions of stability for the ver-
tical point with respect to the possible formation of a different fluid
mass E.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 465
If the system can be in equilibrium when the vertical point has
been replaced by a mass E against which the four masses A, B, C, D
abut, being truncated at their vertices, it is evident that E will have
four vertices, at each of which six surfaces of discontinuity meet.
(Thus at one vertex there will be the surfaces formed by A, B, C,
and E.) The tensions of each set of six surfaces (like those of the
six surfaces formed by A, B, C, and D) must therefore be such that
they can be represented by the six edges of a tetrahedron. When
the tensions do not satisfy these relations, there will be no particular
condition of stability for the point about which A, B, C, and D meet,
since if a mass E should be formed, it would distribute itself along
some of the lines or surfaces which meet at the vertical point, and it
is therefore sufficient to consider the stability of these lines and sur-
faces. We shall suppose that the relations mentioned are satisfied.
If we denote by W v the work gained in forming the mass E (of
such size and form as to be in equilibrium) in place of the portions
of the other masses which are suppressed, and by W s the work ex-
pended in forming the new surfaces in place of the old, it may easily
be shown by a method similar to that used on page 459 that
W s =%W y , (637)
whence W s - W Y =iW y ; (638)
also, that when the volume E is small, the equilibrium of E will be
stable or unstable according as W s and W v are negative or positive.
A critical relation for the tensions is that which makes equilibrium
possible for the system of the five masses A, B, C, D, E, when all
the surfaces are plane. The ten tensions may then be represented in
magnitude and direction by the ten distances of five points in space
a, /3 y y y 6 ? s, viz., the tension of A-B and the direction of its normal
by the line a §, etc. The point e will lie within the tetrahedron
formed by the other points. If we write v K for the volume of E, and
v A , v Bj v c , v-q for the volumes of the parts of the other masses which
are suppressed to make room for E, we have evidently
W y =Pk «k — Pa v a— Pb v b -p c v c —pv v-d. (639)
Hence, when all the surfaces are plane, Wy=0, and T1^=0. Now
equilibrium is always possible for a given small value of v % with any
given values of the tensions and of p A , p B , p c , p D . When the tensions
satisfy the critical relation, W s = 0, if p A =p B =p c =p D . But when
y E is small and constant, the value of W s must be independent of p A ,
Pb, Pc, Pd, since the angles of the surfaces are determined by the
tensions and their curvatures may be neglected. Hence, TJ^= 0, and
466 J. W. Gibbs— Equilibrium of Heterogeneous Substances.
W Y = 0, when the critical relation is satisfied and v E small. This
gives
_ v A p A -f v B p B + v c p c + v v p D
P*- ~. (640)
In calculating the ratios of v A , v B , v c , v», v t , we may suppose all the
surfaces to be plane. Then E will have the form of a tetrahedron,
the vertices of which may be called a, b, c, d, (each vertex being
named after the mass which is not found there,) and v A , v B , v c , v D will
be the volumes of the tetrahedra into which it may be divided
by planes passing through its edges and an interior point e. The
volumes of these tetrahedra are proportional to those of the five
tetrahedra of the figure a fi y d e, as will easily appear if we recollect
that the line ab is common to the surfaces C-D, D-E, E-C, and there-
fore perpendicular to the surface common to the lines y d, d s, e y,
i. e., to the surface y d s, and so in other cases, (it will be observed
that y, d, and e are the letters which do not correspond to a or b) ;
also that the surface a b c is the surface D-E and therefore perpendic-
ular to d e, etc. Let tetr abccl, trian abc, etc. denote the volume of
the tetrahedron or the area of the triangle specified, sin (ab, be),
sin (abc, dbc), sin (abc, ad), etc. the sines of the angles made by the
lines and surfaces specified, and [BODE], [ODEA], etc., the vol-
umes of tetrahedra having edges equal to the tensions of the surfaces
between the masses specified. Then, since we may express the
volume of a tetrahedron either by £ of the product of one side, an edge
leading to the opposite vertex, and the sine of the angle which these
make, or by § of the product of two sides divided by the common
edge and multiplied by the sine of the included angle,
v A : v B : : tetr bede : tetr acde
: : be sin (be, cde) : ac sin (ac, cde)
: : sin (ba, ac) sin (be, cde) : sin (ab, be) sin (ac, cde)
: : sin (yds, fide) sin {ade, aft) : sin {yds, ade) sin {fide, afi)
tetr yfide tetr fiade _ tetr yade tetr afide
trian ftde trian ade ' trian ade trian fide
: : tetr yfide : tetr yade
::[BCDE]:[CDEA].
Hence,
» a :» b :« c :»d::[BCDE]:[CDEA]:[DEAB]:EABC],(641)
and (640) may be written
_ [BCDE]j PA +[CDEA]^ B + [DEAB]^ c +[EABC] i > D (
F% [BCDE]+[CDEA]-f-[DEAB] + [EABC] * V ;
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 467
If the value of p E is less than this, when the tensions satisfy the critical
relation, the point where vertices of the masses A, B, C, D meet is
stable With respect to the formation of any mass of the nature of E.
But if the value of p E is greater, either the masses A, B, C, D cannot
meet at a point in equilibrium, or the equilibrium will be at least
practically unstable.
When the tensions of the new surfaces are too small to satisfy the
critical relation with the other tensions, these surfaces will be con-
vex toward E; when their tensions are too great for that relation,
the surfaces will be concave toward E. In the first case, W Y is
negative, and the equilibrium of the five masses A, B, C, D, E
is stable, but the equilibrium of the four masses A, B, C, D meeting
at a point is impossible or at least practically unstable. This is sub-
ject to the limitation that when p E is sufficiently small the mass E
which will form will be so small that it may be neglected. This will
only be the case when p E is smaller — in general considerably smaller —
than the second number of (642). In the second case, the equilibrium
of the five masses A, B, C, D, E will be unstable, but the equilibrium
of the four masses A, B, C, D will be stable unless v E (calculated for
the case of the five masses) is of insensible magnitude. This will
only be the case when p K is greater — in general considerably greater —
than the second member of (642).
Liquid Films.
When a fluid exists in the form of a thin film between other fluids,
the great inequality of its extension in different directions will give
rise to certain peculiar properties, even when its thickness is sufficient
for its interior to have the properties of matter in mass. The fre-
quent occurrence of such films, and the remarkable properties which
they exhibit, entitle them to particular consideration. To fix our
ideas, we shall suppose that the film is liquid and that the contiguous
fluids are gaseous. The reader will observe our results are not
dependent, so far as their general character is concerned, upon this
supposition.
Let us imagine the film to be divided by surfaces perpendicular to
its sides into small portions of which all the dimensions are of the
same order of magnitude as the thickness of the film, — such portions
to be called elements of the film, — it is evident that far less time will
in general be required for the attainment of approximate equilibrium
between the different parts of any such element and the other fluids
which are immediately contiguous, than for the attainment of equi-
468 J.W. Gibbs — Equilibrium of Heterogeneous Substances.
librium between all the different elements of the film. There will
accordingly be a time, commencing shortly after the formation of the
film, in which its separate elements may be regarded as satisfying the
conditions of internal equilibrium, and of equilibrium with the con-
tiguous gases, while they may not satisfy all the conditions of equi-
librium with each other. It is when the changes due to this want ol
complete equilibrium take place so slowly that the film appears to be
at rest, except so far as it accommodates itself to any change in the
external conditions to which it is subjected, that the characteristic
properties of the film are most striking and most sharply defined.
Let us therefore consider the properties which will belong to a film
sufficiently thick for its interior to have the properties of matter in
mass, in virtue of the approximate equilibrium of all its elements
taken separately, when the matter contained in each element is
regarded as invariable, with the exception of certain substances
which are components of the contiguous gas-masses and have their
potentials thereby determined. The occurrence of a film which pre-
cisely satisfies these conditions may be exceptional, but the discus-
sion of this somewhat ideal case will enable us to understand the
principal laws which determine the behavior of liquid films in
general.
Let us first consider the properties which will belong to each ele-
ment of the film under the conditions mentioned. Let us suppose
the element extended, while the temperature and the potentials
which are determined by the contiguous gas-masses are unchanged.
If the film has no components except those of which the potentials
are maintained constant, there will be no variation of tension in its
surfaces. The same will be true when the film has only one com-
ponent of which the potential is not maintained constant, provided
that this is a component of the interior of the film and not of its sur-
face alone. If we regard the thickness of the film as determined by
dividing surfaces which make the surface-density of this compo-
nent vanish, the thickness will vary inversely as the area of the ele-
ment of the film, but no change will be produced in the nature or
the tension of its surfaces. If, however, the single component of
which the potential is not maintained constant is confined to the sur-
faces of the film, an extension of the element will generally produce
a decrease in the potential of this component, and an increase of ten-
sion. This will certainly be true in those cases in which the compo-
nent shows a tendency to distribute itself with a uniform superficial
density.
J. W. Gribbs — Equilibrium of Heterogeneous Substances. 469
When the film has two or more components of which the potentials
are not maintained constant by the contiguous gas masses, they will
not in general exist in the same proportion in the interior of the film as
on its surfaces, but those components which diminish the tensions will
be found in greater proportion on the surfaces. When the film is ex-
tended, there will therefore not be enough of these substances to keep
up the same volume- and surface-densities as before, and the deficiency
will cause a certain increase of tension. The value of the elasticity of
the flm, (i. e„ the infinitesimal increase of the united tensions of its
surfaces divided by the infinitesimal increase of area in a unit of sur-
face), may be calculated from the quantities which specify the nature
of the film, when the fundamental equations of the interior mass, of
the contiguous gas-masses, and of the two surfaces of discontinuity
are known. We may illustrate this by a simple example.
Let us suppose that the two surfaces of a plane film are entirely
alike, that the contiguous gas-masses are identical in phase, and that
they determine the potentials of all the components of the film
except two. Let us call these components S t and S 2 , the latter
denoting that which occurs in greater proportion on the surface than
in the interior of the film. Let us denote by y 1 and y 2 the densities
of these components in the interior of the film, by A. the thickness of
the film determined by such dividing surfaces as make the surface-
density of S 1 vanish (see page 397), by r s(tl) the surface-density of
the other component as determined by the same surfaces, by <3 and s
the tension and area of one of these surfaces, and by i^the elasticity
of the film when extended under the supposition that the total quan-
tities of $ t and S 2 in the part of the film extended are invariable, as
also the temperature and the potentials of the other components.
From the definition of JS we have
2d6 — E-, (643)
s
and from the conditions of the extension of the film
ds_ d{Xy x ) __ d{ly 2 + 2r 2(1) )
s Xy x ^ij+2r 2(I)
(644)
Hence we obtain
X y x — = — y x dX — Xdy x ,
ds
(Xy 2 +2r 2{1) )- = -y 2 dX-Xdy 2 -2dr^ ) ',
and eliminating c?A,
470 J. W. Gibbs— -Equilibrium of Heterogeneous Substances.
ds
2 yi r 2(i)-~=-^yi<Zy 2 + hy 2 d ri -2y 1 dr 2(iy (645)
If we set r = Ta
(646)
Yx
we have dr = ?* ^ *£* d ^\ (647)
Yx
and
ds
2r 2(D~= — hy 1 dr — 2dr 2(iy (648)
s
With this equation we may eliminate ds from (643). We may also
eliminate do' by the necessary relation [see (514)]
dff = — r 2il) dM 2 .
This will give
4 r 2(1) 2 dju 2 — E (A y x dr + 2 tfr 2(1) ), (649)
or
4/; (1) 2 , <&• di\ a) . :
where the differential coefficients are to be determined on the condi-
tions that the temperature and all the potentials except ju x and // 2
are constant, and that the pressure in the interior of the film shall
remain equal to that in the contiguous gas-masses. The latter con-
dition may be expressed by the equation
(Yi-Yi') d'Mi + (Yz-Yz) dfA 2 — 0, (651)
in which y t ' and y 2 ' denote the densities of S x and S 2 in the con-
tiguous gas-masses. [See (98).] When the tension of the surfaces
of the film and the pressures in its interior and in the contiguous gas-
masses are known in terms of the temperature and potentials, equa-
tion (650) will give the value of E in terms of the same variables
together with A.
If we write G x and G 2 for the total quantities of / S l and S 2 per
unit of area of the film, we have
'^i—^Yi, (652)
G 2 =\y 2 -\-2F 2{ ^ (653)
Therefore,
G z =G 1 r+2r 2ii) ,
iXy fr+tdTp! (664)
* 2 / or, dju 2 dpi 2 '
where the differential coefficients in the second member are to be
determined as in (650), and that in the first member with the addi-
tional condition that G , is constant. Therefore,
\dpL 2 /G l
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 471
+ I\ { ^__/dG 2
E \dfA 2 jG x '
and E=L±rs (l 2( C lth\ , (655)
\dG 2 / g,
the last differential coefficient being determined by the same condi-
tions as that in the preceding equation. It will be observed that the
value of E will be positive in any ordinary case.
These equations give the elasticity of any element of the film when
the temperature and the potentials for the substances which are found
in the contiguous gas-masses are regarded as constant, and the poten-
tials for the other components, /j 1 and jj 2 , have had time to equalize
themselves throughout the element considered. The increase of
tension immediately after a rapid extension will be greater than that
given by these equations.
The existence of this elasticity, which has thus been established
from a priori considerations, is clearly indicated by the phenomena
which liquid films present. Yet it is not to be demonstrated simply
by comparing the tensions of films of different thickness, even when
they are made from the same liquid, for difference of thickness does
not necessarily involve any difference of tension. When the phases
within the films as well as without are the same, and the surfaces of
the films are also the same, there will be no difference of tension.
Nor will the tension of the same film be altered, if a part of the inte-
rior drains away in the course of time, without affecting the surfaces.
In case the thickness of the film is reduced by evaporation, the tension
may be either increased or diminished. (The evaporation of the sub-
stance S 1 , in the case we have just considered, would diminish the
tension.) Yet it may easily be shown that extension increases the
tension of a film and contraction diminishes it. When a plane film
is held vertically, the tension of the upper portions must evidently
be greater than that of the lower. The tensions in every part of the
film may be reduced to equality by turning it into a horizontal posi-
tion. By restoring the original position we may restore the original
tensions, or nearly so. It is evident that the same element of the
film is capable of supporting very unequal tensions. ]STor can this be
always attributed to viscosity of the film. For in many cases, if we
hold the film nearly horizontal, and elevate first one side and then an
other, the lighter portions of the film will dart from one side to the
other, so as to show a very striking mobility in the film. The differ-
ences of tension which cause these rapid movements are only a very
Trans. Conn. Acad., Vol, III. 60 March, 18?8.
472 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
small fraction of the difference of tension in the upper and lower
portions of the film when held vertically.
If we account for the power of an element of the film to support an
increase of tension by viscosity, it will be necessary to suppose that
the viscosity offers a resistance to a deformation of the film in which
its surface is enlarged and its thickness diminished, which is enor-
mously great in comparison with the resistance to a deformation in
which the film is extended in the direction of one tangent and con-
tracted in the direction of another, while its thickness and the areas
of its surfaces remain constant. This is not to be readily admitted
as a physical explanation, although to a certain extent the phenomena
resemble those which would be caused by such a singular viscosity.
(See page 439.) The only natural explanation of the phenomena is
that the extension of an element of the film, which is the immediate
result of an increase of external force applied to its perimeter, causes
an increase of its tension, by which it is brought into true equilibrium
with the external forces.
The phenomena to which we have referred are such as are apparent
to a very cursory observation. In the following experiment, which
is described by M. Plateau,* an increased tension is manifested in a
film while contracting after a previous extension. The warmth of a
finger brought near to a bubble of soap-water with glycerine, which
is thin enough to show colors, causes a spot to appear indicating
a diminution of thickness. When the finger is removed, the spot
returns to its original color. This indicates a contraction, which
would be resisted by any viscosity of the film, and can only be due
to an excess of tension in the portion stretched on the return of its
original temperature.
We have so far supposed that the film is thick enough for its inte-
rior to have the properties of matter in mass. Its properties are then
entirely determined by those of the three phases and the two surfaces
of discontinuity. From these we can also determine, in part at least,
the properties of a film at the limit at which the interior ceases to
have the properties of matter in mass. The elasticity of the film,
which increases with its thinness, cannot of course vanish at that
limit, so that the film cannot become unstable with respect to exten-
sion and contraction of its elements immediately after passing that
limit.
Yet a certain kind of instability will probably arise, which we may
* " Statique experimentale et theorique des liquides soumis aux seules forces mole-
culaires," vol. i, p. 294.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 473
here notice, although it relates to changes in which the condition of
the invariability of the quantities of certain components in an element
of the film is not satisfied. With respect to variations in the distri-
bution of its components, a film will in general be stable, when its
interior has the properties of matter in mass, with the single exception
of variations affecting its thickness without any change of phase or of
the nature of the surfaces. With respect to this kind of change, which
may be brought about by a current in the interior of the film, the
equilibrium is neutral. But when the interior ceases to have the pro-
perties of matter in mass, it is to be supposed that the equilibrium
will generally become unstable in this respect. For it is not likely
that the neutral equilibrium will be unaffected by such a change of
circumstances, and since the film certainly becomes unstable when it
is sufficiently reduced in thickness, it is most natural to suppose that
the first effect of diminishing the thickness will be in the direction of
instability rather than in that of stability. (We are here considering
liquid films between gaseous masses. In certain other cases, the
opposite supposition might be more natural, as in respect to a film of
water between mercury and air, which would certainly become stable
when sufficiently reduced in thickness.)
Let us now return to our former suppositions — that the film is thick
enough for the interior to have the properties of matter in mass, and
that the matter in each element is invariable, except with respect to
those substances which have their potentials determined by the con-
tiguous gas-masses — and consider what conditions are necessary for
equilibrium in such a case.
In consequence of the supposed equilibrium of its several elements,
such a film may be treated as a simple surface of discontinuity
between the contiguous gas-masses (which may be similar or different),
whenever its radius of curvature is very large in comparison with its
thickness, — a condition which we shall always suppose to be fulfilled.
With respect to the film considered in this light, the mechanical
conditions of equilibrium will always be satisfied, or very nearly so,
as soon as a state of approximate rest is attained, except in those
cases in which the film exhibits a decided viscosity. That is, the
relations (613), (614), (615) will hold true, when by a we understand
the tension of the film regarded as a simple surface of discontinuity
(this is equivalent to the sum of the tensions of the two surfaces of
the film), and by F its mass per unit of area diminished by the mass
of gas which would occupy the same space if the film should be sup-
pressed and the gases should meet at its surface of tension. This
f
474 J. W. Gibbs— Equilibrium of Heterogeneous Substances.
surface of tension of the film will evidently divide the distance
between the surfaces of tension for the two surfaces of the film taken
separately, in the inverse ratio of their tensions. For practical pur-
poses, we may regard r simply as the mass of the film per unit of
area. It will be observed that the terms containing Tin (613) and
(614) are not to be neglected in our present application of these
equations.
But the mechanical conditions of equilibrium for the film regarded
as an approximately homogeneous mass in the form of a thin sheet
bounded by two surfaces of discontinuity are not necessarily satisfied
when the film is in a state of apparent rest. In fact, these conditions
cannot be satisfied (in any place where the force of gravity has an
appreciable intensity) unless the film is horizontal. For the pressure
in the interior of the film cannot satisfy simultaneously condition
(612), which requires it to vary rapidly with the height z, and condi-
tion (613) applied separately to the different surfaces, which makes it
a certain mean between the pressures in the adjacent gas-masses.
Nor can these conditions be deduced from the general condition
of mechanical equilibrium (606) or (611), without supposing that the
interior of the film is free to move independently of the surfaces,
which is contrary to what we have supposed.
Moreover, the potentials of the various components of the film will
not in general satisfy conditions (617), and cannot (when the tem-
perature is uniform) unless the film is horizontal. For if these condi-
tions were satisfied, equation (612) would follow as a consequence.
(See page 449.)
We may here remark that such a film as we are considering cannot
form any exception to the principle indicated on page 450,— that
when a surface of discontinuity which satisfies the conditions of
mechanical equilibrium has only one component which is not found
in the contiguous masses, and these masses satisfy all the conditions
of equilibrium, the potential for the component mentioned must satisfy
the law expressed in (617), as a consequence of the condition of
mechanical equilibrium (614). Therefore, as we have just seen that
it is impossible that all the potentials in a liquid film which is not hori-
zontal should conform to (617) when the temperature is uniform, it
follows that if a liquid film exhibits any persistence which is not due
to viscosity, or to a horizontal position, or to differences of tempera-
ture, it must have more than one component of which the potential
is not determined by the contiguous gas-masses in accordance with
(617).
J. W. G-ibbs — Equilibriutn of Heterogeneous Substances. 475
The difficulties of the quantitative experimental verification of the
properties which have been described would be very great, even in
cases in which the conditions we have imagined were entirely ful-
filled. Yet the general effect of any divergence from these condi-
tions will be easily perceived, and when allowance is made for such
divergence, the general behavior of liquid films will be seen to agree
with the requirements of theory.
The formation of a liquid film takes place most symmetrically
when a bubble of air rises to the top of a mass of the liquid. The
motion of the liquid, as it is displaced by the bubble, is evidently
such as to stretch the two surfaces in which the liquid meets the air,
where these surfaces approach one another. This will cause an
increase of tension, which will tend to restrain the extension of the
surfaces. The extent to which this effect is produced will vary with
the nature of the liquid. Let us suppose that the case is one in
which the liquid contains one or more components which, although
constituting but a very small part of its mass, greatly reduce its ten-
sion. Such components will exist in excess on the surfaces of the
liquid. In this case the restraint upon the extension of the surfaces
will be considerable, and as the bubble of air rises above the general
level of the liquid, the motion of the latter will consist largely of a
running out from between the two surfaces. But this running out of
the liquid will be greatly retarded by its viscosity as soon as it is
reduced to the thickness of a film, and the effect of the extension of
the surfaces in increasing their .tension will become greater and
more permanent as the quantity of liquid diminishes which is avail-
able for supplying the substances which go to form the increased sur-
faces.
We may form a rough estimate of the amount of motion which is
possible for the interior of a liquid film, relatively to its exterior, by
calculating the descent of water between parallel vertical planes at
which the motion of the water is reduced to zero. If we use the
coefficient of viscosity as determined by Helmholtz and Piotrowski,*
we obtain
V=58lZ> 9 ~, (656)
where V denotes the mean velocity of the water (i. e., that velocity
* Sitzungsberichte der Wiener Akademie, (mathemat.-naturwiss. Olasse), B. xl, S.
607. The calculation of formula (656) and that of the factor (f ) applied to the formula
of Poiseuille, to adapt it to a current between plane surfaces, have been made by
means of the general equations of the motion of a viscous liquid as given in the
memoir referred to.
476 J. W. Oibbs — Equilibrium of Heterogeneous Substances.
which, if it were uniform throughout the whole space between the
fixed planes, would give the same discharge of water as the actual
variable velocity) expressed in millimetres per second, and D denotes
the distance in millimetres between the fixed planes, which is sup-
posed to be very small in proportion to their other dimensions. This
is for the temperature of 24.5° C. For the same temperature, the
experiments of Poiseuille * give
V- 337 Z> 2
for the descent of water in long capillary tubes, which is equivalent to
F=899Z> 2 (657)
for descent between parallel planes. The numerical coefficient in this
equation differs considerably from that in (656), which is derived from
experiments of an entirely different nature, but we may at least con-
clude that in a film of a liquid which has a viscosity and specific
gravity not very different from those of water at the temperature
mentioned the mean velocity of the interior relatively to the surfaces
will not probably exceed 1000 _Z) 2 . This is a velocity of .l mm per
second for a thickness of .01 mm , .06 mm per minute for a thickness of
.001 (which corresponds to the red of the fifth order in a film of
water), and .036 mm per hour for a thickness of .0001 mm (which corre-
sponds to the white of the first order). Such an internal current is
evidently consistent with great persistence of the film, especially in
those cases in which the film can exist in a state of the greatest
tenuity. On the other hand, the above equations give so large a
value of T^for thicknesses of l n,m or .l mm , that the film can evidently
be formed without carrying up any great weight of liquid, and any
such thicknesses as these can have only a momentary existence.
A little consideration will show that the phenomenon is essentially
of the same nature when films are formed in any other way, as by
dipping a ring or the mouth of a cup in the liquid and then with-
drawing it. When the film is formed in the mouth of a pipe, it may
sometimes be extended so as to form a large bubble. Since the elas-
ticity (i. e., the increase of the tension with extension) is greater in
the thinner parts, the thicker parts will be most extended, and the
effect of this process (so far as it is not modified by gravity) will be
to diminish the ratio of the greatest to the least thickness of the film.
During this extension, as well as at other times, the increased elas-
ticity due to imperfect communication of heat, etc., will serve to pro-
tect the bubble from fracture by shocks received from the air or the
* Ibid., p. 653 ; or Memoires des Savants Strangers, vol. ix, p. 532.
J. W. Gibbs — Equilibrium of Heterogeneous. Substances. 477
pipe. If the bubble is now laid upon a suitable support, the condi-
tion (613) will be realized almost instantly. The bubble will then
tend toward conformity with condition (614), the lighter portions ris-
ing to the top, more or less slowly, according to the viscosity of the
film. The resulting difference of thickness between the upper and
the lower parts of the bubble is due partly to the greater tension to
which the upper parts are subject, and partly to a difference in the
matter of which they are composed. When the film has only two
components of which the potentials are not determined by the con-
tiguous atmosphere, the laws which govern the arrangement of the
elements of the film may be very simply expressed. If we call these
components iS i and S 2i the latter denoting (as on page 469) that
which exists in excess at the surface, one element of the film will
tend toward the same level with another, or a higher, or a lower
level, according as the quantity of S 2 bears the same ratio to the
quantity of iS 1 in the first element as in the second, or a greater, or a
less ratio.
When a film, however formed, satisfies both the conditions (613)
and (614), its thickness being sufficient for its interior to have the
properties of matter in mass, the interior will still be subject to the
slow current which we have already described, if it is truly fluid, how-
ever great its viscosity may be. It seems probable, however, that
this process is often totally arrested by a certain gelatinous consist-
ency of the mass in question, in virtue of which, although practically
fluid in its behavior with reference to ordinary stresses, it may have
the properties of a solid with respect to such very small stresses as
those which are caused by gravity in the interior of a very thin film
which satisfies the conditions (613) and (614).
However this may be, there is another cause which is often more
potent in producing changes in a film, when the conditions just men-
tioned are approximately satisfied, than the action of gravity on its
interior. This will be seen if we turn our attention to the edge
where the film is terminated. At such an edge we generally find a
liquid mass, continuous in phase with the interior of the film, which
is bounded by concave surfaces, and in which the pressure is therefore
less than in the interior of the film. This liquid mass therefore
exerts a strong suction upon the interior of the film, by which its
thickness is rapidly reduced. This effefct is best seen when a film
which has been formed in a ring is held in a vertical position. Unless
the film is very viscous, its diminished thickness near the edge causes
a rapid upward current on each side, while the central portion slowly
478 ./. W. Gibbs — Equilibrium of Heterogeneous Substances.
descends. Also at the bottom of the film, where the edge is nearly-
horizontal, portions which have become thinned escape from their
position of unstable equilibrium beneath heavier portions, and pass
upwards, traversing the central portion of the film until they find a
position of stable equilibrium. By these processes, the whole film is
rapidly reduced in thickness.
The energy of the suction which produces these effects may be
inferred from the following considerations. The pressure in the
slender liquid mass which encircles the film is of course variable,
being greater in the lower portions than in the upper, but it is every-
where less than the pressure of the atmosphere. Let us take a point
where the pressure is less than that of the atmosphere by an amount
represented by a column of the liquid one centimetre in height. (It
is probable that much greater differences of pressure occur.) At a
point near by in the interior of the film the pressure is that of the
atmosphere. Now if the difference of pressure of these two points
were distributed uniformly through the space of one centimetre, the
intensity of its action would be exactly equal to that of gravity.
But since the change of pressure must take place very suddenly (in
a small fraction of a millimetre), its effect in producing a current in a
limited space must be enormously great compared with that of
gravity.
Since the process just described is connected with the descent of
the liquid in the mass encircling the film, we may regard it as
another example of the downward tendency of the interior of the
film. There is a third way in which this descent may take place,
when the principal component of the interior is volatile, viz.,
through the air. Thus, in the case of a film of soap-water, if we
suppose the atmosphere to be of such humidity that the potential for
water at a level mid- way between the top and bottom of the film has
the same value in the atmosphere as in the film, it may easily be
shown that evaporation will take place in the upper portions and
condensation in the lower. These processes, if the atmosphere were
otherwise undisturbed, would occasion currents of diffusion and other
currents, the general effect of which would be to carry the moisture
downward. Such a precise adjustment would be hardly attainable,
and the processes described would not be so rapid as to have a prac-
tical importance.
But when the potential for water in the atmosphere differs con-
siderably from that in the film, as in the case of a film of soap-water
in a dry atmosphere, or a film of soap-water with glycerine in a moist
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 479
atmosphere, the effect of evaporation or condensation is not to be
neglected. In the first case, the diminution of the thickness of the
film will be accelerated, in the second, retarded. In the case of the
film containing glycerine, it should be observed that the water con-
densed cannot in all respects replace the fluid carried down by the
internal current but that the two processes together will tend to
wash out the glycerine from the film.
But when a component which greatly diminishes the tension of the
film, although forming but a small fraction of its mass, (therefore
existing in excess at the surface,) is volatile, the effect of evaporation
and condensation may be considerable, even when the mean value of
the potential for that component is the same in the film as in the sur-
rounding atmosphere. To illustrate this, let us take the simple case
of two components S x and 8 2 , as before. (See page 469.) It appears
from equation (508) that the potentials must vary in the film with
the height z, since the tension does, and from (98) that these varia-
tions must (very nearly) satisfy the relation
y 1 and y 2 denoting the densities of /S 1 and $ 2 in the interior of the
film. The variation of the potential of S 2 as we pass from one level
to another is therefore as much more rapid than that of /S 1 , as its
density in the interior of the film is less. If then the resistances
restraining the evaporation, transmission through the atmosphere,
and condensation of the two substances are the same, these processes
will go on much more rapidly with respect to S 2 . It will be
observed that the values of -— * and --— will have opposite signs,
• dz dz
the tendency of S t being to pass down through the atmosphere, and
that of S 2 to pass up. Moreover, it may easily be shown that the
evaporation or condensation of S 2 will produce a very much greater
effect than the evaporation or condensation of the same quantity of
S v These effects are really of the same kind. For if condensation
of £ 3 takes place at the top of the film, it will cause a diminution of
tension, and thus occasion an extension of this part of the film, by
which its thickness will be reduced, as it would be by evaporation of
S v We may infer that it is a general condition of the persistence of
liquid films, that the substance which causes the diminution of tension
in the upper parts of the film must not be volatile.
But apart from any action of the atmosphere, we have seen that a
Trans. Conn. Acad., Yol. III. 61 April, 1878.
48C J. W. Gibbs — Equilibrium >■ of Heterogeneous Substances.
film which is truly fluid in its interior is in general subject to a con-
tinual diminution of thickness by the internal currents due to gravity
and the suction at its edge. Sooner or later, the interior will some-
where cease to have the properties of matter in mass. The film will
then probably become unstable with respect to a flux of the interior
(see page 473), the thinnest parts tending to become still more thin
(apart from any external cause) very much as if there were an
attraction between the surfaces of the film, insensible at greater dis-
tances, but becoming sensible when the thickness of the film is suffi-
ciently reduced. We should expect this to determine the rupture of
the film, and such is doubtless the case with most liquids. In a film
of soap-water, however, the rupture does not take place, and the
processes which go on can be watched. It is apparent even to a very
superficial observation that a film of which the tint is approaching
the black exhibits a remarkable instability. The continuous change
of tint is interrupted by the breaking out and rapid extension of
black spots. That in the formation of these bright spots a separa-
tion of different substances takes place, and not simply an extension
of a part of the film, is shown by the fact that the film is made
thicker at the edge of these spots.
This is very distinctly seen in a plane vertical film, when a single
black spot breaks out and spreads rapidly over a considerable area
which was before of a nearly uniform tint approaching the black.
The edge of the black spot as it spreads is marked as it were by a
string of bright beads, which unite together on touching, and thus
becoming larger, glide down across the bands of color below. Under
favorable circumstances, there is often quite a shower of these bright
spots. They are evidently small spots very much thicker — appar-
ently many times thicker — than the part of the film out of which
they are formed. Now if the formation of the black spots were due
to a simple extension of the film, it is evident that no such appear-
ance would be presented. The thickening of the edge of the film
cannot be accounted for by contraction. For an extension of the
upper portion of the film and contraction of the lower and thicker
portion, with descent of the intervening portions, would be far less
resisted by viscosity, and far more favored by gravity than such
extensions and contractions as would produce the appearances
described. But the rapid formation of a thin spot by an internal
current would cause an accumulation at the edge of the spot of the
material forming the interior of the film, and necessitate a thickening
of the film in that place.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 481
That which is most difficult to account for in the formation of the
black spots is the arrest of the process by which the film grows thin-
ner. It seems most natural to account for this, if possible, by passive
resistance to motion due to a very viscous or gelatinous condition of
the film. For it does not seem likely that the film, after becoming
unstable by the flux of matter from its interior, would become stable
(without the support of such resistance) by a continuance of the
same process. On the other hand, gelatinous properties are very
marked in soap-water which contains somewhat more soap than is
best for the formation of films, and it is entirely natural that, even
when such properties are wanting in the interior of a mass or thick
film of a liquid, they may still exist in the immediate vicinity of the
surface (where we know that the soap or some of its components
exists in excess), or throughout a film which is so thin that the
interior has ceased to have the properties of matter in mass.* But
these considerations do not amount to any a priori probability of an
arrest of the tendency toward an internal current between adjacent
elements of a black spot which may differ slightly in thickness, in
time to prevent rupture of the film. For, in a thick film, the increase
of the tension with the extension, which is necessary for its stability
with respect to extension, is connected with an excess of the
soap (or of some of its components) at the surface as compared with
the interior of the film. With respect to the black spots, although
the interior has ceased to have the properties of matter in mass, and
any quantitative determinations derived from the surfaces of a mass
of the liquid will not be applicable, it is natural to account for the
stability with reference to extension by supposing that the same
general difference of composition still exists. If therefore we account
for the arrest of internal currents by the increasing density of
soap or some of its components in the interior of the film, we must
still suppose that the characteristic difference of composition in the
interior and surface of the film has not been obliterated.
The preceding discussion relates to liquid films between masses of
gas. Similar considerations will apply to liquid films between other
liquids or between a liquid and a gas, and to films of gas between
* The experiments of M. Plateau (chapter VII of the work already cited) show that
this is the case to a very remarkable degree with respect to a solution of saponine.
With respect to soap-water, however, they do not indicate any greater superficial
viscosity than belongs to pure water. But the resistance to an internal current, such as
we are considering, is not necessarily measured by the resistance to such motions
as those of the experiments referred to.
482 J.W. Gibbs— Equilibrium of Heterogeneous Substances.
masses of liquid. The latter may be formed by gently depositing a
liquid drop upon the surface of a mass of the same or a different
liquid. This may be done (with suitable liquids) so that the con-
tinuity of the air separating the liquid drop and mass is not broken,
but a film of air is formed, which, if the liquids are similar, is a
counterpart of the liquid film which is formed by a bubble of air ris-
ing to the. top of a mass of the liquid. (If the bubble has the same
volume as the drop, the films will have precisely the same form, as
well as the rest of the surfaces which bound the bubble and the
drop.) Sometimes, when the weight and momentum of the drop
carry it through the surface of the mass on which it falls, it appears
surrounded by a complete spherical film of air, which is the counter-
part on a small scale of a soap-bubble hovering in air.* Since, how-
ever, the substance to which the necessary differences of tension in
the film are mainly due is a component of the liquid masses on each
side of the air film, the necessary differences of the potential of this
substance cannot be permanently maintained, and these films have
little persistence compared with films of soap-water in air. In this
respect, the case of these air-films is analogous to that of liquid films
in an atmosphere containing substances by which their tension is
greatly reduced. Compare page 479.
Surfaces of Discontinuity between Solids and Fluids.
We have hitherto treated of surfaces of discontinuity on the sup-
position that the contiguous masses are fluid. This is by far the
most simple case for any rigorous treatment, since the masses are
necessarily isotropic both in nature and in their state of strain. In
this case, moreover, the mobility of the masses allows a satisfactory
experimental verification of the mechanical conditions of equilibrium.
On the other hand, the rigidity of solids is in general so great, that
any tendency of the surfaces of discontinuity to variation in area or
form may be neglected in comparison with the forces which are pro-
duced in the interior of the solids by any sensible strains, so that it
is not generally necessary to take account of the surfaces of discon-
tinuity in determining the state of strain of solid masses. But we
must take account of the nature of the surfaces of discontinuity
* These spherical air-films are easily formed in soap-water. They are distinguish-
able from ordinary air-bubbles by their general behavior and by their appearance.
The two concentric spherical surfaces are distinctly seen, the diameter of one appear-
ing to be about three-quarters as large as that of the other. This is of course an
optical illusion, depending upou the index of refraction of the liquid.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 483
between solids and fluids with reference to the tendency toward
solidification or dissolution at such surfaces, and also with reference to
the tendencies of different fluids to spread over the surfaces of solids.
Let us therefore consider a surface of discontinuity between a fluid
and a solid, the latter being either isotropic or of a continuous crystal-
line structure, and subject to any kind of stress compatible with a
state of mechanical equilibrium with the fluid. We shall not exclude
the case in which substances foreign to the contiguous masses are
present in small quantities at the surface of discontinuity, but we
shall suppose that the nature of this surface (i. <?., of the non-homo-
geneous film between the approximately homogeneous masses), is
entirely determined by the nature and state of the masses which it
separates, and the quantities of the foreign substances which may be
present. The notions of the dividing surface, and of the superficial
densities of energy, entropy, and the several components, which we
have used with respect to surfaces of discontinuity between fluids
(see pages 380 and 386), will evidently apply without modification to
the present case. We shall use the suffix a with reference to the
substance of the solid, and shall suppose the dividing surface to be
determined so as to make the superficial density of this substance
vanish. The superficial densities of energy, of entropy, and of the
other component substances may then be denoted by our usual sym-
bols (see page 39*7),
f S(l)5 7 /s(D? J- 2(1)5 1 3(1)5 e t°-
Let the quantity 6 be defined by the equation
s — £ s (1 )-^/s (1) -^ 2 r 2(l) -/i 3 r 3(1) -etc, (659)
in which t denotes the temperature, and ju 2 , ju 3 , etc. the potentials
for the substances specified at the surface of discontinuity.
As in the case of two fluid masses, (see page 421,) we may regard
6 as expressing the work spent in forming a unit of the surface
of discontinuity — under certain conditions, which we need not here
specify — but it cannot properly be regarded as expressing the tension
of the surface. The latter quantity depends upon the work spent in
stretching the surface, while the quantity 6 depends upon the work
spent informing the surface. With respect to perfectly fluid masses,
these processes are not distinguishable, unless the surface of discon-
tinuity has components which are not found in the contiguous masses,
and even in this case, (since the surface must be supposed to be formed
out of matter supplied at the same potentials which belong to the mat-
ter in the surface,) the work spent in increasing the surface infinitesi-
484 J. W. Gibbs— Equilibrium of Heterogeneous Substances.
mally by stretching is identical with that which must be spent in
forming an equal infinitesimal amount of new surface. But when one
of the masses is solid, and its states of strain are to be distinguished,
there is no such equivalence between the stretching of the surface
and the forming of new surface.*
With these preliminary notions, we now proceed to discuss the
condition of equilibrium which relates to the dissolving of a solid at
the surface where it meets a fluid, when the thermal and mechanical
conditions of equilibrium are satisfied. It will be necessary for us to
consider the case of isotropic and of crystallized bodies separately,
since in the former the value of 6 is independent of the direction of
the surface, except so far as it may be influenced by the state of strain
of the solid, while in the latter the value of a varies greatly with the
direction of the surface with respect to the axes of crystallization, and
in such a manner as to have a large number of sharply defined
minima, f This may be inferred from the phenomena which crystal-
line bodies present, as will appear more distinctly in the following
discussion. Accordingly, while a variation in the direction of an
* This will appear more distinctly if we consider a particular case. Let us consider
a thin plane sheet of a crystal in a vacuum (which may be regarded as a limiting case
of a very attenuated fluid), and let us suppose that the two surfaces of the sheet are
alike. By applying the proper forces to the edges of the sheet, we can make all stress
vanish in its interior. The tensions of the two surfaces, are in equilibrium with these
forces, and are measured by them. But the tensions of the surfaces, thus determined,
may evidently have different values in different directions, and are entirely different
from the quantity which we denote by a, which represents the work required to form
a unit of the surface by any reversible process, and is not connected with any idea of
direction.
In certain cases, however, it appears probable that the values of o and of the
superficial tension will not greatly differ. This is especially true of the numerous
bodies which, although generally (and for many purposes properly) regarded as solids
are really very viscous fluids. Even when a body exhibits no fluid properties at its
actual temperature, if its surface has been formed at a higher temperature, at which
the body was fluid, and the change from the fluid to the solid state has been by
insensible gradations, we may suppose that the value of a coincided with the super-
ficial tension until the body was decidedly solid, and that they will only differ so far
as they may be differently affected by subsequent variations of temperature and of the
stresses applied to the solid. Moreover, when an amorphous solid is in a state of
equilibrium with a solvent, although it may have no fluid properties in its interior, it
seems not improbable that the particles at its surface, which have a greater degree of
mobility, may so arrange themselves that the value of a will coincide with the super-
ficial tension, as in the case of fluids.
f The differential coefficients of a with respect to the direction-cosines of the surface
appear to be discontinuous functions of the latter quantities.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 485
element of the surface may be neglected (with respect to its effect on
the value of a) in the case of isotropic solids, it is quite otherwise
with crystals. Also, while the surfaces of equilibrium between fluids
and soluble isotropic solids are without discontinuities of direction,
being in general curved, a crystal in a state of equilibrium with a
fluid in which it can dissolve is bounded in general by a broken sur-
face consisting of sensibly plane portions.
For isotropic solids, the conditions of equilibrium may be deduced
as follows. If we suppose that the solid is unchanged, except that an
infinitesimal portion is dissolved at the surface where it meets the
fluid, and that the fluid is considerable in quantity and remains
homogeneous, the increment of energy in the vicinity of the surface
will be represented by the expression
/K- V 4- (e x + c 2 ) e s(1) ] 6NDs
where Ds denotes an element of the surface, dJV the variation in its
position (measured normally, and regarded as negative when the solid
is dissolved), c x and e 2 its principal curvatures (positive when
their centers lie on the same side as the solid), <? S(1) the surface-
density of energy, e y ' and e v " the volume-densities of energy in the
solid and fluid respectively, and the sign of integration relates to the
elements Ds. In like manner, the increments of entropy and of the
quantities of the several components in the vicinity of the surface
will be
J'W-ih" + (<>i+«.) %(d] wi>s,
/[-rZ+i^+^r^WDs,
etc.
The entropy and the matter of different kinds represented by these
expressions we may suppose to be derived from the fluid mass.
These expressions, therefore, with a change of sign, will represent
the increments of entropy and of the quantities of the components
in the whole space occupied by the fluid except that which
is immediately contiguous to the solid. Since this space may be
regarded as constant, the increment of energy in this space may be
obtained [according to equation (12)] by multiplying the above
expression relating to entropy by —t, and those relating to the
components by — yu/, -yu 2 , etc.,* and taking the sum. If to this
* The potential fi , " is marked by double accents in order to indicate that its value
is to be determined in the fluid mass, and to distinguish it from the potential /n , '
486 J. W. Gibbs— Equilibrium of Heterogeneous Substances.
we acid the above expression for the increment of energy near the
surface, we obtain the increment of energy for the whole system.
Now by (93) we have
p" = _ Ey " + t i h " + Ml ' Yl " + M2 y 2 » + etc.
By this equation and (059), our expression for the total increment of
energy in the system may be reduced to the form
/ |V — t r) v ' — J u 1 " ri '+p f ' + (e 1 +c 2 ) or] SiVDs. (660)
In order that this shall vanish for any values of dN, it is necessary
that the coefficient of SJVDs shall vanish. This gives for the condi-
tion of equilibrium
Mi — —, . (661)
This equation is identical with (387), with the exception of the term
containing a, which vanishes when the surface is plane.*
We may also observe that when the solid has no stresses except an
isotropic pressure, if the quantity represented by a is equal to the true
tension of the surface, p" + (c 1 -f c 2 ) o' will represent the pressure in
the interior of the solid, and the second member of the equation will
represent [see equation (93)] the value of the potential in the solid
for the substance of which it consists. In this case, therefore, the
equation reduces to
that is, it expresses the equality of the potentials for the substance of
the solid in the two masses — the same condition which would subsist
if both masses were fluid.
Moreover, the compressibility of all solids is so small that, although
& may not represent the true tension of the surface, nor jt/-|- (e 1 -\-c 2 ) a
the true pressure in the solid when its stresses are isotropic, the quan-
tities s x ' and rj v ' if calculated for the pressure p" -f- {c x -\- e 2 ) a with
the actual temperature will have sensibly the same values as if calcu-
lated for the true pressure of the solid. Hence, the second member
relating to the solid mass (when this is in a state of isotropic stress), which, as we
shall see, may not always have the same value. The other potentials /n. 2 , etc., have
the same values as in (659), and consist of two classes, one of which relates to sub-
stances which are components of the fluid mass, (these might be marked by the double
accents.) and the other relates to substances found only at the surface of discontinuity.
The expressions to be multiplied by the potentials of this latter class all have the
value zero.
* In equation (38*7), the density of the solid is denoted by T, which is therefore
equivalent to y,' in (661).
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 487
of equation (661), when the stresses of the solid are sensibly iso-
tropic, is sensibly equal to the potential of the same body at the
same temperature but with the pressure p" -\- (c x -f- c 2 ) o~, and the
condition of equilibrium with respect to dissolving for a solid of
isotropic stresses may be expressed with sufficient accuracy by saying
that the potential for the substance of the solid in the fluid must
have this value. In like manner, when the solid is not in a state of
isotropic stress, the difference of the two pressures in question will
not sensibly affect the values of e y ' and 7/ v ', and the value of the
second member of the equation may be calculated as if p"-\- {c 1 + c 2 ) G
represented the true pressure in the solid in the direction of the nor-
mal to the surface. Therefore, if we had taken for granted that the
quantity 6 represents the tension of a surface between a solid and a
fluid, as it does when both masses are fluid, this assumption would
not have led us into any practical error in determining the value of
the potential /u," which is necessary for equilibrium. On the other
hand, if in the case of any amorphous body the value of 6 differs
notably from the true surface-tension, the latter quantity substituted
for 6 in (661) will make the second member of the equation equal to
the true value of ///, when the stresses are isotropic, but this will not
be equal to the value of yu/ in case of equilibrium, unless c t ~{-c 2 := 0.
When the stresses in the solid are not isotropic, equation (661)
may be regarded as expressing the condition of equilibrium with
respect to the dissolving of the solid, and is to be distinguished from
the condition of equilibrium with respect to an increase of solid
matter, since the new matter would doubtless be deposited in a state
of isotropic stress. (The case would of course be different with
crystalline bodies, which are not considered here.) The value of
yu/' necessary for equilibrium with respect to the formation of new
matter is a little less than that necessary for equilibrium with respect
to the dissolving of the solid. In regard to the actual behavior of
the solid and fluid, all that the theory enables us to predict with
certainty is that the solid will not dissolve if the value of the poten-
tial jj. t " is greater than that given by the equation for the solid with
its distorting stresses, and that new matter will not be formed if the
value of jX x " is less than the same equation would give for the case of
the solid with isotropic stresses.* It seems probable, however, that
* The possibility that the new solid matter might differ in composition from the
original solid is here left out of account. This point has been discussed on pages
134-137, but without reference to the state of strain of the solid or the influence of
the curvature of the surface of discontinuity. The statement made above may be
Trans. Conn. Acad., Yol. III. 62 April, 1818.
488 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
if the fluid in contact with the solid is not renewed, the system will
generally find a state of equilibrium in which the outermost portion
of the solid will be in a state of isotropic stress. If at first the solid
should dissolve, this would supersaturate the fluid, perhaps until a state
is reached satisfying the condition of equilibrium with the stressed
solid, and then, if not before, a deposition of solid matter in a state of
isotropic stress would be likely to commence and go on until the fluid
is reduced to a state of equilibrium with this new solid matter.
The action of gravity will not affect the nature of the condition of
equilibrium for any single point at which the fluid meets the solid,
but it will cause the values of p" and yu/ in (661) to vary according
to the laws expressed by (612) and (61 1). If we suppose that the
outer part of the solid is in a state of isotropic stress, which is the
most important case, since it is the only one in which the equilibrium
is in every sense stable, we have seen that the condition (661) is at
least sensibly equivalent to this : — that the potential for the sub-
stance of the solid which would belong to the solid mass at the
temperature t and the pressure p"-{- (c 1 -\- c 2 ) 6 must be equal to ///'.
Or, if we denote by (p') the pressure belonging to solid with the
temperature t and the potential equal to ju^', the condition may be
expressed in the form
(p')=p"+(c 1 +e 2 )ff. (662)
Now if we write y" for the total density of the fluid, we have by (612)
dp"=-gy"dz.
By (98) d(p') = y 1 'dj.i 1 ",
and by (617) djj. x ' = — g dz;
whence d ( p') =: — g y x ' dz.
Accordingly we have
d(p')-dp» = g(y»- ri ')dz,
and
(p')-p"=g(y"-y 1 ')z,
z being measured from the horizontal plane for which (p') —p".
Substituting this value in (662), we obtain
C l+ C 2= ^ % ( 663 )
generalized so as to hold true of the formation of new solid matter of any kind on
the surface as follows : — that new solid matter of any kind will not be formed upon
the surface (with more than insensible thickness), if the second member of (661) cal-
culated for such new matter is greater than the potential in the fluid for such matter.
J. W. Gibbs — Equilibrium of Heterogeneous Substcmces. 489
precisely as if both masses were fluid, and a denoted the tension of
their common surface, and {p) the true pressure in the mass specified.
[Compare (619). J
The obstacles to an exact experimental realization of these rela-
tions are very great, principally from the want of absolute uniformity
in the internal structure of amorphous solids, and on account of the
passive resistances to the processes which are necessary to bring
about a state satisfying the conditions of theoretical equilibrium,
but it may be easy to verify the general tendency toward diminution
of surface, which is implied in the foregoing equations.*
Let us apply the same method to the case in which the solid is
a crystal. The surface between the solid and fluid will now consist
of plane portions, the directions of which may be regarded as invari-
* It seems probable that a tendency of this kind plays an important part in some
of the phenomena which have been observed with respect to the freezing together
of pieces of ice. (See especially Professor Faraday's " Note on Kegelation" in the
Proceedings of the Royal Society, vol. x, p. 440 ; or in the Philosophical Magazine, 4th ser.,
vol. xxi, p. 146.) Although this is a body of crystalline structure, and the action
which takes place is doubtless influenced to a certain extent by the directions of
the axes of crystallization, yet, since the phenomena have not been observed to
depend upon the orientation of the pieces of ice, we may conclude that the effect, so
far as its general character is concerned, is such as might take place with an isotropic
body. In other words, for the purposes of a general explanation of the phenomena
we may neglect the differences in the values of cyw (the suffixes are used to indicate
that the symbol relates to the surface between ice and water) for different orientations
of the axes of crystallization, and also neglect the influence of the surface of discon-
tinuity with respect to crystalline structure, which must be formed by the freezing
together of the two masses of ice when the axes of crystallization in the two masses
are not similarly directed. In reality, this surface — or the necessity of the formation
of such a surface if the pieces of ice freeze together — must exert an influence adverse
to their union, measured by a quantity an, which is determined for this surface by
the same principles as when one of two contiguous masses is fluid, and varies with
the orientations of the two systems of crystallographic axes relatively to each other
and to the surface. But under the circumstances of the experiment, since we may
neglect the possibility of the two systems of axes having precisely the same directions,
this influence is probably of a tolerably constant character, and is evidently not suffi-
cient to alter the general nature of the result. In order wholly to prevent the
tendencj'- of pieces of ice to freeze together, when meeting in water with curved sur-
faces and without pressure, it would be necessary that o"rr— 2cr IW) except so far as the
case is modified by passive resistances to change, and by the inequality in the values
of on and a nv for different directions of the axes of crystallization.
It will be observed that this view of the phenomena is in harmony with the
opinion of Professor Faraday. With respect to the union of pieces of ice as an
indirect consequence of pressure, see page 198 of volume xi of the Proceedings of the
Royal Society ; or the Philosophical Magazine, 4th ser., vol. xxiii, p. 407.
490 J. W. Gibbs— Equilibrium of Heterogeneous Substances.
able. If the crystal grows on one side a distance dJV, without other
change, the increment of energy in the vicinity of the surface will be
Ov'-O s 6JV+ ^'(fsd/ V cosec co'—s sw V cot go') SJV,
where e v ' and s v " denote the volume-densities of energy in the crystal
and fluid respectively, s the area of the side on which the crystal
grows, f S(1) the surface-density of energy on that side, e S(l) ' the surface-
density of energy on an adjacent side, go' the external angle of these
two sides, I' their common edge, and the symbol 2' a summation
with respect to the different sides adjacent to the first. The incre-
ments of entropy and of the quantities of the several components will
be represented by analogous formulas, and if we deduce as on pages 485,
486 the expression for the increase of energy in the whole system due
to the growth of the crystal without change of the total entropy or
volume, and set this expression equal to zero, we shall obtain for the
condition of equilibrium
(V— tr t v'-^"Y i ' J r p")sSJSr
+ 2' ( a' I' cosec go' — a V cot go') 6A t — 0, (664)
where 6 and a' relate respectively to the same sides as f S(1) and £ S(1) ' in
the preceding formula. This gives
s v ' — 1 7/ v ' +p" ^'( g' V cosec go' — 6 V cot go')
ft" = r -^+ -* ^ '• («65)
It will be observed that unless the side especially considered is
small or narrow, we may neglect the second fraction in this equation,
which will then give the same value of /.//' as equation (38V), or as
equation (661) applied to a plane surface.
Since a similar equation must hold true with respect to every other
side of the crystal of which the equilibrium is not affected by meet-
ing some other body, the condition of equilibrium for the crystalline
form (when unaffected by gravity) is that the expression
2'(ff' I' cosec go'— o' I' cot go') ,„„„*
— ^ '- (666)
shall have the same value for each side of the crystal. (By the value
of this expression for any side of the crystal is meant its value when
6 and 8 are determined by that side and the other quantities by the
surrounding sides in succession in connection with the first side.)
This condition will not be affected by a change in the size of a crys-
tal while its proportions remain the same. But the tendencies of
similar crystals toward the form required by this condition, as mea-
sured by the inequalities in the composition or the temperature of
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 491
the surrounding fluid which would counterbalance them, will be
inversely as the linear dimensions of the crystals, as appears from the
preceding equation.
If we write v for the volume of a crystal, and 2(ff«) for the sum
of the areas of all its sides multiplied each by the corresponding
value of <7, the numerator and denominator of the fraction (666),
multiplied each by dN, may be represented by 62(o~s) and dv
respectively. The value of the fraction is therefore equal to that of
the differential coefficient
d2(o's)
dv
as determined by the displacement of a particular side while the other
sides are fixed. The condition of equilibrium for the form of a crys-
tal (when the influence of gravity may be neglected) is that the
value of this differential coefficient must be independent of the partic-
ular side which is supposed to be displaced. For a constant volume
of the crystal, 2(0 s) has therefore a minimum value when the
condition of equilibrium is satisfied, as may easily be proved more
directly.
When there are no foreign substances at the surfaces of the crystal,
and the surrounding fluid is indefinitely extended, the quantity
2(cr s) represents the work required to form the surfaces of the
crystal, and the coefficient of s SJ^in (664) with its sign reversed rep-
resents the work gained in forming a mass of volume unity like the
crystal but regarded as without surfaces. We may denote the work
required to form the crystal by
W S -W Y ,
W$ denoting the work required to form the surfaces [i. e., 2(0s)],
and Wy the work gained in forming the mass as distinguished from
the surfaces. Equation (664) may then be written
-dW Y + 2(0ds) = O. (667)
Now (664) would evidently continue to hold true if the crystal were
diminished in size, remaining similar to itself in form and in nature,
if the values of o" in all the sides were supposed to diminish in the
same ratio as the linear dimensions of the crystal. The variation of
W s would then be determined by the relation
d W s = d2(0 s) = 1 2(0 ds),
and that of W v by (667). Hence,
dW & =z§dW v ,
492 J.W. Gibbs — Equilibrium of Heterogeneous Substances.
and, since T^ and W v vanish together,
W s - W v = iW s = iW Y , (668)
— the same relation which we have before seen to subsist with respect
to a spherical mass of fluid as well as in other cases. (See pages 421,
425, 465.)
The equilibrium of the crystal is unstable with respect to variations
in size when the surrounding fluid is indefinitely extended, but it
may be made stable by limiting the quantity of the fluid.
To take account of the influence of gravity, we must give to }A t "
and p" in (665) their average values in the side considered. These
coincide (when the fluid is in a state of internal equilibrium) with
their values at the center of gravity of the side. The values of
Ti'j f v'j Vv may be regarded as constant, so far as the influence of
gravity is concerned. Now since by (612) and (617)
dp"=-gy"dz,
and
dfj-i' = — g dz,
we have
d(y l 'M 1 "-p") = g(y"-y 1 ')dz.
Comparing (664), we see that the upper or the lower faces of the
crystal will have the greater tendency to grow, (other things being
equal,) according as the crystal is lighter or heavier than the fluid.
When the densities of the two masses are equal, the effect of gravity
on the form of the crystal may be neglected.
In the preceding paragraph the fluid is regarded as in a state of
internal equilibrium. If we suppose the composition and tempera-
ture of the fluid to be uniform, the condition which will make the
effect of gravity vanish will be that
~dT '
when the value of the differential coefficient is determined in accord-
ance with this supposition. This condition reduces to
\ dp )t,<m //'
which, by equation (92), is equivalent to
(-*)" =-L- (66 9)
* A suffixed m is used to represent all the symbols m x , m. 2 , etc., except such as
may occur in the differential coefficient.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 493
The tendency of a crystal to grow will be greater in the upper or
lower parts of the fluid, according as the growth of a crystal at con-
stant temperature and pressure will produce expansion or contraction.
Again, we may suppose the composition of the fluid and its
entropy per unit of mass to be uniform. The temperature will then
vary with the pressure, that is, with z. We may also suppose the
temperature of different crystals or different parts of the same crystal
to be determined by the fluid in contact with them. These condi-
tions express a state which may perhaps be realized when the fluid is
gently stirred. Owing to the differences of temperature we cannot
regard t>' and ij Y ' in (664) as constant, but we may regard their
variations as subject to the relation de Y ' =. t dr/ v '. Therefore, if we
make ;/ v ' = for the mean temperature of the fluid, (which involves
no real loss of generality,) we may treat e v ' — t ;/ v ' as constant. We
shall then have for the condition that the effect of gravity shall
vanish —
d(y i > l "-p")_
dz '
which signifies in the present case that
\ dp Jrj,m Ki"
or, by (90),
(*)" =i, (670)
Since the entropy of the crystal is zero, this equation expresses that
the dissolving of a small crystal in a considerable quantity of the
fluid will produce neither expansion nor contraction, when the pres-
sure is maintained constant and no heat is supplied or taken away.
The manner in which crystals actually grow or dissolve is often
principally determined by other differences of phase in the surround-
ing fluid than those which have been considered in the preceding
paragraph. This is especially the case when the crystal is growing
or dissolving rapidly. When the great mass of the fluid is consider-
ably supersaturated, the action of the crystal keeps the part immedi-
ately contiguous to it nearer the state of exact saturation. The
farthest projecting parts of the crystal will therefore be most exposed
to the action of the supersaturated fluid, and will grow most rapidly.
The same parts of a crystal will dissolve most rapidly in a fluid con-
siderably below saturation.*
* See 0. Lehmann "Ueber dasWachsthum der Krystalle," Zeitschrift fur Krystal-
lographie imd Mineralogie, Bd. i, S. 453 ; or the review of the paper in Wiedemann's
Beiblatter, Bd. ii, S. 1.
494 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
But even when the fluid is supersaturated only so much as is
necessary in order that the crystal shall grow at all, it is not to be
expected that the form in which 2(o's) has a minimum value (or
such a modification of that form as may be due to gravity or to the
influence of the body supporting the crystal) will always be the
ultimate result. For we cannot imagine a body of the internal
structure and external form of a crystal to grow or dissolve by an
entirely continuous process, or by a process in the same sense continu-
ous as condensation or evaporation between a liquid and gas, or the
corresponding processes between an amorphous solid and a fluid.
The process is rather to be regarded as periodic, and the formula
(664) cannot properly represent the true value of the quantities
intended unless 6JV is equal to the distance between two successive
layers of molecules in the crystal, or a multiple of that distance.
Since this can hardly be treated as an infinitesimal, we can only con-
clude with certainty that sensible changes cannot take place for
which the expression (664) would have a positive value.*
* That it is necessary that certain relations shall be precisely satisfied in order that
equilibrium may subsist between a liquid and gas with respect to evaporation, is
explained (see Clausius " Ueber die Art der Bewegung, welche wir Warme nennen,"
Pogg. Ann., Bd. c, S. 353 ; or Abhand. iiber die mech. Wiirmetheorie, XIV,) by suppos-
ing that a passage of individual molecules from the one mass to the other is continually
taking place, so that the slightest circumstance may give the preponderance to the
passage of matter in either direction. The same supposition may be applied, at least
in many cases, to the equilibrium between amorphous solids and fluids. Also in the
case of crystals in equilibrium with fluids, there may be a passage of individual mole-
cules from one mass to the other, so as to cause insensible fluctuations in the mass of
the solid. If these fluctuations are such as to cause the occasional deposit or removal
of a whole layer of particles, the least cause would be sufficient to make the probability
of one kind of change prevail over that of the other, and it would be necessary for
equilibrium that the theoretical conditions deduced above should be precisely satisfied.
But this supposition seems quite improbable, except with respect to a very small side.
The following view of the molecular state of a crystal when in equilibrium with
respect to growth or dissolution appears as probable as any. Since the molecules at
the corners and edges of a perfect crystal would be less firmly held in their places
than those in the middle of a side, we may suppose that when the condition of
theoretical equilibrium (665) is satisfied several of the outermost layers of molecules
on each side of the crystal are incomplete toward the edges. The boundaries of these
imperfect layers probably fluctuate, as individual molecules attach themselves to the
crystal or detach themselves, but not so that a layer is entirely removed (on any side
of considerable size), to be restored again simply by the irregularities of the motions
of the individual molecules. Single molecules or small groups of molecules may
indeed attach themselves to the side of the crystal but they will speedily be dislodged,
and if any molecules are thrown out from the middle of a surface, these deficiencies
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 495
Let us now examine the special condition of equilibrium which
relates to a line at which three different masses meet, when one or
more of these masses is solid. If we apply the method of page 685
to a system containing such a line, it is evident that we shall obtain
in the expression corresponding to (660), beside the integral relating
to the surfaces, a term of the form
to be interpreted as the similar term in (611), except so far as the
definition of 6 has been modified in its extension to solid masses. In
order that this term shall be incapable of a negative value it is neces-
will also soon be made good ; nor will the frequency of these occurrences be such as
greatly to affect the general smoothness of the surfaces, except near the edges where
the surfaces fall off somewhat, as before described. Now a continued growth on any
side of a crystal is impossible unless new layers can be formed. This will require a
value of /V' which may exceed that given by equation (665) by a finite quantity.
Since the difficulty in the formation of a new layer is at or near the commencement
of the formation, the necessary value of \i x " may be independent of the area of the
side, except when the side is very small. The value of (i , " which is necessary for the
growth of the crystal will however be different for different kinds of surfaces, and
probably will generally be greatest for the surfaces for which a is least.
On the whole, it seems not improbable that the form of very minute crystals in
equilibrium with solvents is principally determined by equation (665), (i. e., by the
condition that 2(c s) shall be a minimum for the volume of the crystal except so far as
the case is modified by gravity or the contact of other bodies,) but as they grow
larger (in a solvent no more supersaturated than is necessary to make them grow at
all), the deposition of new matter on the different surfaces will be determined more by
the nature (orientation) of the surfaces and less by their size and relations to the
surrounding surfaces. As a final result, a large crystal, thus formed, will generally
be bounded by those surfaces alone on which the deposit of new matter takes place
least readily, with small, perhaps insensible truncations. If one kind of surfaces
satisfying this condition cannot form a closed figure, the crystal will be bounded by
two or three kinds of surfaces determined by the same condition. The kinds of
surface thus determined will probably generally be those for which a has the least
values. But the relative development of the different kinds of sides, even if unmodi-
fied by gravity or the contact of other bodies, will not be such as to make 2(crs) a
minimum. The growth of the crystal will finally be confined to sides of a single kind.
It does not appear that any part of the operation of removing a layer of molecules
presents any especial difficulty so marked as that of commencing a new layer ; yet
the values of fi , " which will just allow the different stages of the process to go on
must be slightly different, and therefore, for the continued dissolving of the crystal
the value of ft t " must be less (by a finite quantity) than that given by equation (665).
It seems probable that this would be especially true of those sides for which a has
the least values. The effect of dissolving a crystal (even when it is done as slowly
as possible) is therefore to produce a form which probably differs from that of
theoretical equilibrium in a direction opposite to that of a growing crystal.
Trans. Conn. Acad., Vol. III. 63 June, 1818.
496 J. W. Gibbs— Equilibrium of Heterogeneous Substances.
sary that at every point of the line
2{<j6T)^0 (671)
for any possible displacement of the line. Those displacements are to
be regarded as possible which are not prevented by the solidity of
the masses, when the interior of every solid mass is regarded as
incapable of motion. At the surfaces between solid and fluid masses,
the processes of solidification and dissolution will be possible in some
cases, and impossible in others.
The simplest case is when two masses are fluid and the third is
solid and insoluble. Let us denote the solid by S, the fluids by
A and B, and the angles filled by these fluids by a and /3 respec-
tively. If the surface of the solid is continuous at the line where it
meets the two fluids, the condition of equilibrium reduces to
o- AB cos a= ff BS ~ cr AS . (672)
If the line where these masses meet is at an edge of the solid, the
condition of equilibrium is that
(T AB COSaSff BS -ff AS , 1
and ^abCOS/?^0- as -0- bs ; [ (b73)
which reduces to the preceding when a+fi—n. Since the dis-
placement of the line can take place by a purely mechanical process,
this condition is capable of a more satisfactory experimental verifica-
tion than those conditions which relate to processes of solidification
and dissolution. Yet the frictional resistance to a displacement of
the line is enormously greater than in the case of three fluids,
since the relative displacements of contiguous portions of matter are
enormously greater. Moreover, foreign substances adhering to the
solid are not easily displaced, and cannot be distributed by extensions
and contractions of the surface of discontinuity, as in the case of
fluid masses. Hence, the distribution of such substances is arbitrary
to a greater extent than in the case of fluid masses, (in which a single
foreign substance in any surface of discontinuity is uniformly distri-
buted, and a greater number are at least so distributed as to make the
tension of the surface uniform,) and the presence of these substances
will modify the conditions of equilibrium in a more irregular manner.
If one or more of three surfaces of discontinuity which meet in a
line divides an amorphous solid from a fluid in which it is soluble
such a surface is to be regarded as movable, and the particular condi-
tions involved in (671) will be accordingly modified. If the soluble
solid is a crystal, the case will properly be treated by the method
used on page 490. The condition of equilibrium relating to the line
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 497
will not in this case be entirely separable from those relating to the
adjacent surfaces, since a displacement of the line will involve a dis-
placement of the whole side of the crystal which is terminated at this
line. But the expression for the total increment of energy in the
system due to any internal changes not involving any variation in
the total entropy or volume will consist of two parts, of which one
relates to the properties of the masses of the system, and the other
may be expressed in the form
62{0s),
the summation relating to all the surfaces of discontinuity. This
indicates the same tendency toward changes diminishing the value of
2(o~ s), which appears in other cases.*
General Relations. — For any constant state of strain of the surface
of the solid, we may write
de S{1) =. ■■tdTfa l) + /* 2 dr a{1) + tA a dr sm + etc., (674)
since this relation is implied in the definition of the quantities
involved. From this and (659) we obtain
da — - T?ato)dt-r 2W dp 9 .-r 3il) dpi 3 - etc., (675)
which is subject, in strictness, to the same limitation — that the state
of strain of the surface of the solid remains the same. But this
limitation may in most cases be neglected. (If the quantity 6 repre-
sented the true tension of the surface, as in the case of a surface
between fluids, the limitation would be wholly unnecessary.)
Another method and notation, — We have so far supposed that we
have to do with a non-homogeneous film of matter between two
homogeneous (or very nearly homogeneous) masses, and that the
nature and state of this film is in all respects determined by the
* The freezing together of wool and ice may be mentioned here. The fact that
a fiber of wool which remains in contact with a block of ice under water will become
attached to it seems to be strictly analogous to the fact that if a solid body be brought
into such a position that it just touches the free surface of water, the water will
generally rise up about the point of contact so as to touch the solid over a surface of
some extent. The condition of the latter phenomenon is
0S A +%A> 0SW7
where the suffixes s, a, and w refer to the solid, to air, and to water, respectively. In
like manner, the condition for the freezing of the ice to the wool, if we neglect
the seolotropic properties of the ice, is
ffsw +0iw> ff si»
where, the suffixes s , w , and i relate to wool, to water, and to ice, respectively. See
Proc. Roy. Soc, vol. x, p. 447 ; or Phil. Mag., 4th ser., vol. xxi, p. 151.
498 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
nature and state of these masses together with the quantities of the
foreign substances which may be present in the film. (See page 483.)
Problems relating to processes of solidification and dissolution seem
hardly capable of a satisfactory solution, except on this supposition,
which appears in general allowable with respect to the surfaces pro-
duced by these processes. But in considering the equilibrium of
fluids at the surface of an unchangeable solid, such a limitation is
neither necessary nor convenient. The following method of treating
the subject will be found more simple and at the same time more
general.
Let us suppose the superficial density of energy to be determined
by the excess of energy in the vicinity of the surface over that which
would belong to the solid, if (with the same temperature and state
of strain) it were bounded by a vacuum in place of the fluid, and to
the fluid, if it extended with a uniform volume-density of energy just
up to the surface of the solid, or, if in any case this does not suffi-
ciently define a surface, to a surface determined in some definite way
by the exterior particles of the solid. Let us use the symbol (e s ) to
denote the superficial energy thus defined. Let us suppose a superficial
density of entropy to be determined in a manner entirely analogous,
and be denoted by (?/ s ). In like manner also, for all the components
of the fluid, and for all foreign fluid substances which may be present
at the surface, let the superficial densities be determined, and denoted
by (F 2 ), (^"3), etc. These superficial densities of the fluid components
relate solely to the matter which is fluid or movable. All matter
which is immovably attached to the solid mass is to be regarded as a
part of the same. Moreover, let S be defined by the equation
? = (8 s )-t( Vs )-fi 2 (r 2 )-// 3 (r 3 )- etc. (676)
These quantities will satisfy the following general relations —
d(s s ) = t d(r h ) +yu 2 d(r 2 ) + ju 3 d{T z ) + etc -> &11)
ds= — (//s) dt—{F 2 ) d/x 2 — (T 3 ) dfx z — etc. (678)
In strictness, these relations are subject to the same limitation as
(674) and (675). But this limitation may generally be neglected.
In fact, the values of ?, (f s ), etc. must in general be much less
affected by variations in the state of strain of the surface of the solid
than those of o~, £ S(1) , etc.
The quantity s evidently represents the tendency to contraction in
that portion of the surface of the fluid which is in contact with the
solid. It may be called the superficial tension of the fluid in contact
with the solid. Its value may be either positive or negative.
J. Wl Gibbs — Equilibrium, of Heterogeneous Substances. 499
It will be observed that for the same solid surface and for the same
temperature but for different fluids the values of 6 (in all cases to
which the definition of this quantity is applicable) will differ from
those of s by a constant, viz., the value of a for the solid surface in
a vacuum.
For the condition of equilibrium of two different fluids at a line on
the surface of the solid, we may easily obtain
cr AB cos a = ? BS - s AS} (679)
the suffixes, etc., being used as in (6*72), and the condition being
subject to the same modification when the fluids meet at an edge of
the solid.
It must also be regarded as a condition of theoretical equilibrium
at the line considered, [subject, like (679), to limitation on account
of passive resistances to motion,] that if there are any foreign sub-
stances in the surfaces A-S and B-S, the potentials for these sub-
stances shall have the same value on both sides of the line ; or, if
any such substance is found only on one side of the line, that the
potential for that substance must not have a less value on the other
side ; and that the potentials for the components of the mass A, for
example, must have the same values in the surface B-C as in the
mass A, or, if they are not actual components of the surface B-C, a
value not less than in A. Hence, we cannot determine the difference
of the surface-tensions of two fluids in contact with the same solid, by
bringing them together upon the surface of the solid, unless these
conditions are satisfied, as well as those which are necessary to pre-
vent the mixing of the fluid masses.
The investigation on pages 442-448 of the conditions of equilibrium
for a fluid system under the influence of gravity may easily be
extended to the case in which the system is bounded by or includes
solid masses, when these can be treated as rigid and incapable of
dissolution. The general condition of mechanical equilibrium would
be of the form
— fp SDv +fgydzl)v+fff 6Ds -\-fgTdz Ds
+ fgdzI>m + fsdDs+fg(r)dzD8=0, (680)
where the first four integrals relate to the fluid masses and the sur-
faces which divide them, and have the same signification as in
equation (606), the fifth integral relates to the movable solid masses,
and the sixth and seventh to the surfaces between the solids and
fluids, (r) denoting the sum of the quantities (F 2 ), (r z ), etc. It
should be observed that at the surface where a fluid meets a solid
500 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
6z and dz, which indicate respectively the displacements of the solid
and the fluid, may have different values, but the components of
these displacements which are normal to the surface must be equal.
From this equation, among other particular conditions of equilib-
rium, we may derive the following —
ds=g(r)dz, (681)
[compare (614),] which expresses the law governing the distribu-
tion of a thin fluid film on the surface of a solid, when there are no
passive resistances to its motion.
By applying equation (680) to the case of a vertical cylindrical tube
containing two different fluids, we may easily obtain the well-known
theorem that the product of the perimeter of the internal surface by
the difference S r — s" of the superficial tensions of the upper and lower
fluids in contact with the tube is equal to the excess of weight of the
matter in the tube above that which would be there, if the boundary
between the fluids were in the horizontal plane at which their pres-
sures would be equal. In this theorem, we may either include or
exclude the weight of a film of fluid matter adhering to the tube.
The proposition is usually applied to the column of fluid in mass
between the horizontal plane for which p'=jp" and the actual
boundary between the two fluids. The superficial tensions s' and s"
are then to be measured in the vicinity of this column. But we may
also include the weight of a film adhering to the internal surface of
the tube. For example, in the case of water in equilibrium with its
own vapor in a tube, the weight of all the water-substance in the
tube above the plane p'=p", diminished by that of the water-vapor
which would fill the same space, is equal to the perimeter multiplied
by the difference in the values of s at the top of the tube and at the
plane p'=z p". If the height of the tube is infinite, the value of s at
the top vanishes, and the weight of the film of water adhering to the
tube and of the mass of liquid water above the plane p'—p" dimin-
ished by the weight of vapor which would fill the same space is
equal in numerical value but of opposite sign to the product of the
perimeter of the internal surface of the tube multiplied by i", the
superficial tension of liquid water in contact with the tube at the
pressure at which the water and its vapor would be in equilibrium at
a plane surface. In this sense, the total weight of water which can
be supported by the tube per unit of the perimeter of its surface is
directly measured by the value of - s for water in contact with the
tube.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 501
Modification of the conditions of equilibrium by electro-
motive force. — Theory of a perfect electro-chemical
apparatus.
We know by experience that in certain fluids (electrolytic con-
ductors) there is a connection between the fluxes of the component
substances and that of electricity. The quantitative relation between
these fluxes may be expressed by an equation of the form
De = H + etc. _ s h - _ et / 682)
where De, Drn^ etc. denote the infinitesimal quantities of electricity
and of the components of the fluid which pass simultaneously through
any same surface, which may be either at rest or in motion, and
a A , a h , etc., a g , a h , etc. denote positive constants. We may evidently
regard Dm^, Dm h , etc., Dm g , Dm h , etc., as independent of one
another. For, if they were not so, one or more could be expressed
in terms of the others, and we could reduce the equation to a shorter
form in which all the terms of this kind would be independent.
Since the motion of the fluid as a whole will not involve any elec-
trical current, the densities of the components specified by the suf-
fixes must satisfy the relation
-^ + r^ + n + n +
These densities, therefore, are not independently variable, like the
densities of the components which we have employed in other cases.
We may account for the relation (682) by supposing that electric-
ity (positive or negative) is inseparably attached to the different
kinds of molecules, so long as they remain in the interior of the fluid,
in such a way that the quantities a a , a b , etc. of the substances speci-
fied are each charged with a unit of positive electricity, and the quan-
tities a g , a^ etc. of the substances specified by these suffixes are each
charged with a unit of negative electricity. The relation (683) is
accounted for by the fact that the constants a a , a g , etc. are so small
that the electrical charge of any sensible portion of the fluid varying
sensibly from the law expressed in (683) would be enormously great,
so that the formation of such a mass would be resisted by a very
great force.
It will be observed that the choice of the substances which we
regard as the components of the fluid is to some extent arbitrary, and
that the same physical relations may be expressed by different equa-
502 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
tions of the form (682), in which the fluxes are expressed with refer-
ence to different sets of components. If the components chosen are
such as represent what we believe to be the actual molecular consti-
tution of the fluid, those of which the fluxes appear in the equation of
the form (682) are called the ions, and the constants of the equation
are called their electro-chemical equivalents. For our present pur-
pose, which has nothing to do with any theories of molecular consti-
tution, we may choose such a set of components as may be conven-
ient, and call those ions, of which the fluxes appear in the equation of
the form (682), without farther limitation.
Now, since the fluxes of the independently variable components of
an electrolytic fluid do not necessitate any electrical currents, all the
conditions of equilibrium which relate to the movements of these
components will be the same as if the fluid were incapable of the
electrolytic process. Therefore all the conditions of equilibrium which
we have found without reference to electrical considerations, will
apply to an electrolytic fluid and its independently variable compo-
nents. But we have still to seek the remaining conditions of equili-
brium, which relate to the possibility of electrolytic conduction.
For simplicity, we shall suppose that the fluid is without internal
surfaces of discontinuity (and therefore homogeneous except so far as
it may be slightly affected by gravity), and that it meets metallic
conductors {electrodes) in different parts of its surface, being other-
wise bounded by non-conductors. The only electrical currents which
it is necessary to consider are those which enter the electrolyte at
one electrode and leave it at another.
If all the conditions of equilibrium are fulfilled in a given state of
the system, except those which relate to changes involving a flux of
electricity, and we imagine the state of the system to be varied by
the passage from one electrode to another of the quantity of electric-
ity Se accompanied by the quantity dm A of the component specified,
without any flux of the other components or any variation in the
total entropy, the total variation of energy in the system will be rep-
resented by the expression
( F " _ V') Se + (///' - fO 3m» + (2"'— T ") $™»
in which V, V" denote the electrical potentials in pieces of the same
kind of metal connected with the two electrodes, V, T", the gravita-
tional potentials at the two electrodes, and ptj, ptj', the intrinsic
potentials for the substance specified. The first term represents the
increment of the potential energy of electricity, the second the incre-
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 503
ment of the intrinsic energy of the ponderable matter, and the third
the increment of the energy due to gravitation.* But by (682)
6m & = a & 6e
It is therefore necessary for equilibrium that
V" - V + <** (,u a " - pi - T" + V) = 0. (684)
To extend this relation to all the electrodes we may write
V + «. W - V) - V" + a a (//." - T")
= V" -f a a {/jJ" - T">) = etc. (685)
For each of the other cations (specified by b etc.) there will be a sim-
ilar condition, and for each of the anions a condition of the form
V _ a K ( M >- T) = V" - a g ( Ms " - T")
= V" - a g (,u g "' - V") = etc. (686)
When the effect of gravity may be neglected, and there are but
two electrodes, as in a galvanic or electrolytic cell, we have for any
cation
V" - V = ff a (/*.' - yu a "), (687)
and for any anion
V" - V' = a g ( Mg " - Mg >), (688)
where V" — V denotes the electromotive force of the combination.
That is: —
When all the conditions of equilibrium are fulfilled in a galvanic
or electrolytic cell, the electromotive force is equal to the difference in
the values of the potential for any ion or apparent ion at the surfaces
of the electrodes multiplied by the electro-chemical equivalent of that
ion, the greater potential of an anion being at the same electrode as
the greater electrical potential, and the reverse being true of a cation.
Let us apply this principle to different cases.
(I.) If the ion is an independently variable component of an elec-
trode, or by itself constitutes an electrode, the potential for the ion
(in any case of equilibrium which does not depend upon passive resist-
ances to change) will have the same value within the electrode as on
its surface, and will be determined by the composition of the elec-
trode with its temperature and pressure. This might be illustrated
by a cell with electrodes of mercury containing certain quantities of
zinc in solution (or with one such electrode and the other of pure
* It is here supposed that the gravitational potential may be regarded as constant
for each electrode. When this is not the case, the expression may be applied to small
parts of the electrodes taken separately.
Trans. Conn. Acad., Vol. III. 64 June 1878.
5C4 J. W. Gfibbs — Equilibrium of Heterogeneous Substances.
zinc) and an electrolytic fluid containing a salt of zinc, but not capa-
ble of dissolving the mercury.* We may regard a cell in which
hydrogen acts as an ion between electrodes of palladium charged with
hydrogen as another illustration of the same principle, but the solid-
ity of the electrodes and the consequent resistance to the diffusion
of the hydrogen within them (a process which cannot be assisted by
convective currents as in a liquid mass) present considerable obstacles
to the experimental verification of the relation.
(II.) Sometimes the ion is soluble (as an independently variable
component) in the electrolytic fluid. Of course its condition in the
fluid when thus dissolved must be entirely different from its condi-
tion when acting on an ion, in which case its quantity is not inde-
pendently variable, as we have already seen. Its diffusion in the
fluid in this state of solution is not necessarily connected with any
electrical current, and in other relations its properties may be entirely
changed. In any discussion of the internal properties of the fluid
(with respect to its fundamental equation, for example,) it would be
necessary to treat it as a different substance. (See page 117.) But
if the process by which the charge of electricity passes into the
electrode, and the ion is dissolved in the electrolyte is reversible, we
may evidently regard the potentials for the substance of the ion in
(68V) or (688) as relating to the substance thus dissolved in the
electrolyte. In case of absolute equilibrium, the density of the sub-
stance thus dissolved would of course be uniform throughout the
fluid, (since it can move independently of any electrical current,) so
that by the strict application of our principle we only obtain the
somewhat barren result, that if any of the ions are soluble in the fluid
without their electrical charges, the electromotive force must vanish
in any case of absolute equilibrium not dependent upon passive resist-
ances. Nevertheless, cases in which the ion is thus dissolved in the
electrolytic fluid only to a very small extent, and its passage from
one electrode to the other by ordinary diffusion is extremely slow,
may be regarded as approximating to the case in which it is incapable
of diffusion. In such cases, we may regard the relations (687),
(688) as approximately valid, although the condition of equilibrium
* If the electrolytic fluid dissolved the mercury as well as the zinc, equilibrium
could only subsist when the electromotive force is zero, and the composition of the
electrodes identical. For when the electrodes are formed of the two metals in differ-
ent proportions, that which has the greater potential for zinc will have the less poten-
tial for mercury. [See equation (98).] This is inconsistent with equilibrium, accord-
ing to the principle mentioned above, if both metals can act as cations.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 505
relating to the diffusion of the dissolved ion is not satisfied. This
may be the case with hydrogen and oxygen as ions (or apparent ions)
between electrodes of platinum in some of its forms.
(III.) The ion may appear in mass at the electrode. Tf it be a
conductor of electricity, it may be regarded as forming an electrode,
as soon as the deposit has become thick enough to have the proper-
ties of matter in mass. The case therefore will not be different from
that first considered. When the ion is a non-conductor, a continuous
thick deposit on the electrode would of course prevent the possibility
of an electrical current. But the case in which the ion being a non-
conductor is disengaged in masses contiguous to the electrode but
not entirely covering it, is an important one. It may be illustrated
by hydrogen appearing in bubbles at a cathode. In case of perfect
equilibrium, independent of passive resistances, the potential of the
ion in (687) or (688) may be determined in such a mass. Yet the
circumstances are- quite unfavorable for the establishment of perfect
equilibrium, unless the ion is to some extent absorbed by the electrode
or electrolytic fluid, or the electrode is fluid. For if the ion must pass
immediately into the non-conducting mass, while the electricity passes
into the electrode, it is evident that the only possible terminus of an
electrolytic current is at the line where the electrode, the non-conduct-
ing mass, and the electrolytic fluid meet, so that the electrolytic pro-
cess is necessarily greatly retarded, and an approximate ceasing of the
current cannot be regarded as evidence that a state of approximate
equilibrium has been reached. But even a slight degree of solubility
of the ion in the electrolytic fluid or in the electrode may greatly
diminish the resistance to the electrolytic process, and help toward
producing that state of complete equilibrium which is supposed in the
theorem we are discussing. And the mobility of the surface of a
liquid electrode may act in the same way. When the ion is absorbed
by the electrode, or by the electrolytic fluid, the case of course comes
under the heads which we have already considered, yet the fact that
the ion is set free in mass is important, since it is in such a mass that
the determination of the value of the potential will generally be
most easily made.
(IV.) When the ion is not absorbed either by the electrode or by
the electrolytic fluid, and is not set free in mass, it may still be
deposited on the surface of the electrode. Although this can take
place only to a limited extent (without forming a body having the
properties of matter in mass), yet the electro-chemical equivalents of
all substances are so small that a very considerable flux of electricity
506 J.W. Gibbs — Equilibrium of Heterogeneous Substances.
may take place before the deposit will have the properties of matter
in mass. Even when the ion appears in mass, or is absorbed by the
electrode or electrolytic fluid, the non-homogeneous film between the
electrolytic fluid and the electrode may contain an additional portion
of it. Whether the ion is confined to the surface of the electrode
or not, we may regard this as one of the cases in which we have to
recognize a certain superficial density of substances at surfaces of
discontinuity, the general theory of which we have already considered.
The deposit of the ion will affect the superficial tension of the
electrode if it is liquid, or the closely related quantity which we have
denoted by the same symbol o' (see pages 482-500) if the electrode
is solid. The effect can of course be best observed in the case of a
liquid electrode. But whether the electrodes are liquid or solid, if
the external electromotive force V — V" applied to an electrolytic
combination is varied, when it is too weak to produce a lasting current,
and the electrodes are thereby brought into a new state of polariza-
tion, in which they make equilibrium with the altered value of the
electromotive force, without change in the nature of the electrodes or
of the electrolytic fluid, then by (508) or (675)
de"=- r/djj/;
and by (687),
d{ V - V") =~a a (djtj - djdj).
Hence
d( V- V") = ^-,da'- -^ d6". (689)
If we suppose that the state of polarization of only one of the elec-
trodes is affected (as will be the case when its surface is very small
compared with that of the other), we have
do'=^d{V'~V"). (690)
The superficial tension of one of the electrodes is then a function of
the electromotive force.
This principle has been applied by M. Lippmann to the construc-
tion of the electrometer which bears his name.* In applying equa-
tions (689) and (690) to dilute sulphuric acid between electrodes of
mercury, as Jin a Lippmann's electrometer, we may suppose that the
* See his memoir: "Relations entre les phsnomene3 electriques et capillaires."
Annates de Chimieetcle Physique, 5e serie, t. v, p. 494.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 507
suffix refers to hydrogen. It will be most convenient to suppose the
dividing surface to be so placed as to make the surface-density of
mercury zero. (See page 397.) The matter which exists in excess or
deficiency at the surface may then be expressed by the surface-densi-
ties of sulphuric acid, of water, and of hydrogen. The value of the
last may be determined from equation (690). According to M. Lipp-
mann's determinations, it is negative when the surface is in its natural
state (i. e., the state to which it tends when no external electromo-
tive force is applied), since <?' increases with V" — V. When
V" — V is equal to nine-tenths of the electromotive force of a Dan-
iell's cell, the electrode to which V" relates remaining in its natural
state, the tension o' of the surface of the other electrode has a maxi-
mum value, and there is no excess or deficiency of hydrogen at that
surface. This is the condition toward which a surface tends when it
is extended while no flux of electricity takes place. The flux of elec-
tricity per unit of new surface formed, which will maintain a surface
T '
m a constant condition while it is extended, is represented by — -
in numerical value, and its direction, when r a " is negative, is from
the mercury into the acid.
We have so far supposed, in the main, that there are no passive
resistances to change, except such as vanish with the rapidity of the
processes which they resist. The actual condition of things with
respect to passive resistances appears to be nearly as follows. There
does not appear to be any passive resistance to the electrolytic pro-
cess by which an ion is transferred from one electrode to another,
except such as vanishes with the rapidity of the process. For, in any
case of equilibrium, the smallest variation of the externally applied
electromotive force appears to be sufficient to cause a (temporary)
electrolytic current. But the case is not the same with respect to
the molecular changes by which the ion passes into new combinations
or relations, as when it enters into the mass of the electrodes, or sep-
arates itself in mass, or is dissolved (no longer with the properties of
anjon) in the electrolytic fluid. In virtue of the passive resistance to
these processes, the external electromotive force may often vary
within wide limits, without creating any current by which the ion is
transferred from one of the masses considered to the other. In other
words, the value of V - V" may often differ greatly from that
obtained from (687) or (688) when we determine the values of the
potentials for the ion as in cases I, II, and III. We may, however,
regard these equations as entirely valid, when the potentials for the
508 J. Wl G-ibbs — Equilibrium of Heterogeneous Substances.
ions are determined at the surface of the electrodes with reference to
the ion in the condition in which it is brought there or taken away
by an electrolytic current, without any attendant irreversible pro-
cesses. But in a complete discussion of the properties of the surface
of an electrode it may be necessary to distinguish (both in respect to
surface-densities and to potentials) between the substance of the ion
in this condition and the same substance in other conditions into
which it cannot pass (directly) without irreversible processes. No
such distinction, however, is necessary when the substance of the ion
can pass at the surface of the electrode by reversible processes from
any one of the conditions in which it appears to any other.
The formula? (68V), (688) afford as many equations as there are
ions. These, however, amount to only one independent equation
additional to those which relate to the independently variable com-
ponents of the electrolytic fluid. This appears from the considera-
tion that a flux of any cation may be combined with a flux of any
anion in the same direction so as to involve no electrical current, and
that this may be regarded as the flux of an independently variable
component of the electrolytic fluid.
General Properties of a Perfect Electro-chemical Apparatus.
When an electrical current passes through a galvanic or electro-
lytic cell, the state of the cell is altered. If no changes take place in
the cell except during the passage of the current, and all changes
which accompany the current can be reversed by reversing the cur-
rent, the cell may be called a perfect electro-chemical apparatus.
The electromotive force of the cell may be determined by the equa-
tions which have just been given. But some of the general relations
to which such an apparatus is subject may be conveniently stated in
a form in which the ions are not explicitly mentioned.
In the most general case, we may regard the cell as subject to
external action of four different kinds. (1) The supply of electricity
at one electrode and the withdrawal of the same quantity at the
other. (2) The supply or withdrawal of a certain quantity of heat.
(3) The action of gravity. (4) The motion of the surfaces enclosing
the apparatus, as when its volume is increased by the liberation of
gases.
The increase of the energy in the cell is necessarily equal to that
which it receives from external sources. We may express this by the
equation
ds = ( V - V") de + dQ + dW a + dW P , (691)
JT. IV. Gibbs — Equilibrium of Heterogeneous Substances. 509
in which de denotes the increment of the intrinsic energy of the cell,
de the quantity of electricity which passes through it, V and V"
the electrical potentials in masses of the same kind of metal con-
nected with the anode and cathode respectively, dQ the heat received
from external bodies, d W G the work done by gravity, and d W P the
work done by the pressures which act on the external surface of the
apparatus.
The conditions under which we suppose the processes to take place
are such that the increase of the entropy of the apparatus is equal to
the entropy which it receives from external sources. The only exter-
nal source of entropy is the heat which is communicated to the cell
by the surrounding bodies. If we write d?j for the increment of
entropy in the cell, and t for the temperature, we have
drf — -5( (692)
Eliminating dQ, we obtain
ds=(V - V") de + t d V + dW G + dW F , (693)
or
It is worth while to notice that if we give up the condition of the
reversibility of the processes, so that the cell is no longer supposed
to be a perfect electro-chemical apparatus, the relation (691) will still
subsist. But, if we still suppose, for simplicity, that all parts of the
cell have the same temperature, which is necessarily the case with a
perfect electro-chemical apparatus, we shall have, instead of (692),
and instead of (693), (694)
(Y» - V') tie ^ — ds + tdrj + dW G + JWj,. (696)
The values of the several terms of the second member of (694), for
a given cell, will vary with the external influences to which the cell
is subjected. If the^cell is enclosed (with the products of electrolysis)
in a rigid envelop, the last term will vanish. The term relating to
gravity is generally to be neglected. If no heat is supplied or with-
drawn, the term containing drj will vanish. But in the calculation of
the electromotive force, which is the most important application of
the equation, it is generally more convenient to suppose that the tem-
perature remains constant.
510 J. W. G-ibbs — Equilibrium of Heterogeneous Substances.
The quantities expressed by the terms containing d Q and drj in
(691), (69.3), (694), and (696) are frequently neglected in the consid-
eration of cells of which the temperature is supposed to remain con-
stant. In other words, it is frequently assumed that neither heat nor
cold is produced by the passage of an electrical current through a
perfect electro-chemical combination (except that heat which may be
indefinitely diminished by increasing the time in which a given quan-
tity of electricity passes), and that only heat can be produced in any
cell, unless it be by processes of a secondary nature, which are not
immediately or necessarily connected with the process of electrolysis.
It does not appear that this assumption is justified by any sufficient
reason. In fact, it is easy to find a case in which the electromotive
force is determined, entirely by the term t— in (694), all the other
de
terms in the second member of the equation vanishing. This is true
of a Grove's gas battery charged with hydrogen and nitrogen. In
this case, the hydrogen passes over to the nitrogen, — a process which
does not alter the energy of the cell, when maintained at a constant
temperature. The work done by external pressures is evidently
nothing, and that done by gravity is (or may be) nothing. Yet an
electrical current is produced. The work done (or which may be
done) by the current outside of the cell is the equivalent of the work
(or of a part of the work) which might be gained by allowing the
gases to mix in other ways. This is equal, as has been shown by
Lord Rayleigh,* to the work which may be gained by allowing each
gas separately to expand at constant temperature from its initial
volume to the volume occupied by the two gases together. The same
work is equal, as appears from equations (278), (279) on page 217,
(see also page 220,) to the increase of the entropy of the system
multiplied by the temperature.
It is possible to vary the construction of the cell in such a way
that nitrogen or other neutral gas will not be necessary. Let the cell
consist of a U-shaped tube of sufficient height, and have pure hydro-
gen at each pole under very unequal pressures (as of one and two
atmospheres respectively) which are maintained constant by properly
weighted pistons, sliding in the arms of the tube. The difference of
the pressures in the gas-masses at the two electrodes must of course
be balanced by the difference in the height of the two columns of
acidulated water. It will hardly be doubted that such an apparatus
* Philosophical Magazine, vol. xlix, p. 311.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 511
would have an electromotive force acting in the direction of a current
which would carry the hydrogen from the denser to the rarer mass.
Certainly the gas could not be carried in the opposite direction by
an external electromotive force without the expenditure of as much
(electromotive) work as is equal to the mechanical work necessary to
pump the gas from the one arm of the tube to the other. And if by any
modification of the metallic electrodes (which remain unchanged by
the passage of electricity) we could reduce the passive resistances to
zero, so that the hydrogen could be carried reversibly from one mass
to the other without finite variation of the electromotive force, the only
possible value of the electromotive force would be represented by the
expression t-=— , as a very close approximation. It will be observed
that, although gravity plays an essential part in a cell of this kind
by maintaining the difference of pressure in the masses of hydrogen,
the electromotive force cannot possibly be ascribed to gravity, since
the work done by gravity, when hydrogen passes from the denser to
the rarer mass, is negative.
Again, it is entirely improbable that the electrical currents caused
by differences in the concentration of solutions of salts, (as in a cell
containing sulphate of zinc between zinc electrodes, or sulphate of
copper between copper electrodes, the solution of the salt being of
unequal strength at the two electrodes,) which have recently been
investigated theoretically and experimentally by MM. Helmholtz and
Moser,* are confined to cases in which the mixture of solutions of
different degrees of concentration will produce heat. Yet in cases in
which the mixture of more and less concentrated solutions is not
attended with evolution or absorption of heat, the electromotive force
must vanish in a cell of the kind considered, if it is determined
simply by the diminution of energy in the cell. And when the mix-
ture produces cold, the same rule would make any electromotive force
impossible except in the direction which would tend to increase the
difference of concentration. Such conclusions as would be quite
irreconcilable with the theory of the phenomena given by Professor
Helmholtz.
A more striking example of the necessity of taking account of the
variations of entropy in the cell in a priori determinations of electro-
motive force is afforded by electrodes of zinc and mercury in a solu-
tion of sulphate of zinc. Since heat is absorbed when zinc is dissolved
* Annalen der Physik und Cliemie, Neue Folge, Band iii, February, 1878.
Trans. Conn. Acad., Vol. III. 65 Junk, 1878.
512 J. W. Gfibbs — Equilibrium of Heterogeneous Substances.
in mercury,* the energy of the cell is increased by a transfer of zinc
to the mercury, when the temperature is maintained constant. Yet
in this combination, the electromotive force acts in the direction of
the current producing such a transfer. f The couple presents certain
anomalies when a considerable quantity of zinc is united with the
mercury. The electromotive force changes its direction, so that this
case is usually cited as an illustration of the principle that the electro-
motive force is in the direction of the current which diminishes the
energy of the cell, i. e., which produces or allows those changes which
are accompanied by evolution of heat when they take place directly.
But whatever may be the cause of* the electromotive force which has
been observed acting in the direction from the amalgam through the
electrolyte to the zinc (a force which according to the determinations
of M. Gaugain is only one twenty-fifth part of that which acts in the
reverse direction when pure mercury takes the place of the amalgam),
these anomalies can hardly affect the general conclusions with which
alone we are here concerned. If the electrodes of a cell are pure
zinc and an amalgam containing zinc not in excess of the amount
which the mercury will dissolve at the temperature of the experiment
without losing its fluidity, and if the only change (other than thermal)
accompanying a current is a transfer of zinc from one electrode to
the other, — conditions which may not have been satisfied in all the
experiments recorded, but which it is allowable to suppose in a
theoretical discussion, and which certainly will not be regarded as
inconsistent with the fact that heat is absorbed when zinc is dissolved
in mercury, — it is impossible that the electromotive force should be
in the direction of a current transferring zinc from the amalgam to
the electrode of pure zinc. For, since the zinc eliminated from the
amalgam by the electrolytic process might be re-dissolved directly,
such a direction of the electromotive force would involve the pos-
sibility of obtaining an indefinite amount of electromotive work, and
therefore of mechanical work, without other expenditure than that of
heat at the constant temperature of the cell.
None of the cases which we have been considering involve com-
binations by definite proportions, and, except in the case of the cell
with electrodes of mercury and zinc, the electromotive forces are
quite small. It may perhaps be thought that with respect to those
cells in which combinations take place by definite proportions the
electromotive force may be calculated with substantial accuracy from
* J. Eegnauld, Comptes Rendus, t. li, p. 7*78.
f Gaugain, Comptes Rendus, t. xlii, p. 430.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 513
the diminution of the energy, without regarding the variation of
entropy. But the phenomena of chemical combination do not in
general seem to indicate any possibility of obtaining from the com-
bination of substances by any process whatever an amount of mechani-
cal work which is equivalent to the heat produced by the direct union
of the substances.
A kilogramme of hydrogen, for example, combining by combustion
under the pressure of the atmosphere with eight kilogrammes of oxygen
to form liquid water, yields an amount of heat which may be repre-
sented in round numbers by 34000 calories.* We may suppose that
the gases are taken at the temperature of 0° C, and that the water is
reduced to the same temperature. But this heat cannot be obtained
at any temperature desired. A very high temperature has the effect
of preventing to a greater or less extent, the combination of the
elements. Thus, according to M. Sainte-Claire Deville,f the tempera-
ture obtained by the combustion of hydrogen and oxygen cannot
much if at all exceed 2500° C, which implies that less than one-half
of the hydrogen and oxygen present combine at that ternperatm*e.
This relates to combustion under the pressure of the atmosphere.
According to the determinations of Professor BunsenJ in regard
to combustion in a confined space, only one-third of a mixture of
hydrogen and oxygen will form a chemical compound at the tem-
perature of 2850° C. and a pressure of ten atmospheres, and only a
little more than one-half when the temperature is reduced by the
addition of nitrogen to 2024° C, and the pressure to about three
atmospheres exclusive of the part due to the nitrogen.
Now 10 calories at 2500° C. are to be regarded as reversibly con-
vertible into one calorie at 4° C. together with the mechanical work
representing the energy of 9 calories. If, therefore, all the 34000 cal-
ories obtainable from the union of hydrogen and oxygen under atmos-
pheric pressure could be obtained at the temperature of 2500° C, and
no higher, we should estimate the electromotive work performed in a
perfect electro-chemical apparatus in which these elements are com-
bined or separated at ordinary temperatures and under atmospheric
pressure as representing nine-tenths of the 34000 calories, and the
heat evolved or absorbed in the apparatus as representing one-tenth
of the 34000 calories.§ This, of course, would give an electromotive
* See Biihlmann's Handbuch der mechanischen Wwrmetheorie, Bd. ii, p. 290.
f Comptes Bendus, t. lvi, p. 199; and t. lxiv, 61.
% Pogg. Ann., Bd. cxxxi (1867), p. 161.
§ These numbers are not subject to correction for the pressure of the atmosphere,
since the 34000 calories relate to combustion under the same pressure.
514 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
force exactly nine-tenths as great as is obtained on the supposition
that all the 34000 calories are convertible into electromotive or
mechanical work. But, according to all indications, the estimate
2500° C. (for the temperature at which we may regard all the heat of
combustion as obtainable) is far too high,* and we must regard the
theoretical value of the electromotive force necessary to electrolyze
water as considerably less than nine-tenths of the value obtained on
the supposition that it is necessary for the electromotive agent to
supply all the energy necessary for the process.
The case is essentially the same with respect to the electrolysis of
hydrochloric acid, which is probably a more typical example of the
process than the electrolysis of water. The phenomenon of dissocia-
tion is equally marked, and occurs at a much lower temperature, more
than half of the gas being dissociated at 1400° C.f And the heat
which is obtained by the combination of hydrochloric acid gas with
water, especially with water which already contains a considerable
quantity of the acid, is probably only to be obtained at temperatures
comparatively low. This indicates that the theoretical value of the
electromotive force necessary to electrolyze this acid (i. e., the elec-
tromotive force which would be necessary in a reversible electro-
chemical apparatus), must be very much less than that which could
perform in electromotive work the equivalent of all the heat evolved
in the combination of hydrogen, chlorine and water to form the liquid
submitted to electrolysis. This presumption, based upon the phenom-
ena exhibited in the direct combination of the substances, is corrobo-
rated by the experiments of M. Favre, who has observed an absorp-
tion of heat in the cell in which this acid was electrolyzed.J The
* Unless the received ideas concerning the behavior of gases at high temperatures
are quite erroneous, it is possible to indicate the general character of a process
(involving at most only such difficulties as are neglected in theoretical discussions) by
which water may be converted into separate masses of hydrogen and oxygen without
other expenditure than that of an amount of heat equal to the difference of energy of
the matter in the two states and supplied at a temperature far below 2500° C. The
essential parts of the process would be (1) vaporizing the water and heating it to a
temperature at which a considerable part will be dissociated, (2) the pai-tial separation
of the hydrogen and oxygen by filtration, and (3) the cooling of both gaseous masses
until the vapor they contain is condensed. A little calculation will show that in a
continuous process all the heat obtained in the operation of cooling the products of
filtration could be utilized in heating fresh water.
f Sainte-Olaire Deville, Gomptes Rendus, t. lxiv, p. 61.
\ See Memoires des Savants Strangers, Ser. 2, t. xxv, No. 1, p. 1 42 ; or Gomptes Eendus,
t. lxxiii, p. 913. The figures obtained by M. Favre will be given hereafter, in connec-
tion with others of the same nature.
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 515
electromotive work expended must therefore have been less than the
increase of energy in the cell.
In both cases of composition in definite proportions which we have
considered, the compound has more entropy than its elements, and
the difference is by no means inconsiderable. This appears to be the
rule rather than the exception with respect to compounds which have
less energy than their elements. Yet it would be rash to assert that
it is an invariable rule. And when one substance is substituted for
another in a compound, we may expect great diversity in the rela-
tions of energy and entropy.
In some cases, there is a striking correspondence between the elec-
tromotive force of a cell and the rate of diminution of its energy per
unit of electricity transmitted, the temperature remaining constant.
A Daniell's cell is a notable example of this correspondence. It may
perhaps be regarded as a very significant case, since of all cells in
common use, it has the most constant electromotive force, and most
nearly approaches the condition of reversibility. If we apply our
previous notation [compare (691)] with the substitution of finite for
infinitesimal differences to the determinations of M. Favre,* estimat-
ing energy in calories, we have for each equivalent (32.6 kilogrammes)
of zinc dissolved
( V" - V')Ae = 2432V cal -, As — - 25394 caL , A Q = - lQQ^:
It will be observed that the electromotive work performed by the cell
is about four per cent, less than the diminution of energy in the cell.f
The value of A Q, which, when negative, represents the heat evolved
in the cell when the external resistance of the circuit is very great,
was determined by direct measurement, and does not appear to have
been corrected for the resistance of the cell. This correction would
diminish the value of — A Q, and increase that of ( V" - V) Ae, which
was obtained by subtracting — A Q from —As.
It appears that under certain conditions neither heat nor cold is
produced in a Grove's cell. For M. Favre has found that with dif-
ferent degrees of concentration of the nitric acid sometimes heat and
sometimes cold is produced.J When neither is produced, of course
* See Mem, Savants Strang., loc. cit, p. 90 ; or Gomptes Bendus, vol. lxix, p. 35, where
the numbers are slightly different.
•f- A comparison of the experiments of different physicists has in some cases given
a much closer correspondence. See Wiedemann's Galvanismus, etc., 2 te Auflage, Bd.
ii, §§ 1111. 1118.
\ Mem. Savants fitrang., loc. cit., p. 93 ; or Comptes Bendus, t. lxix, p. 3*7, and t.
lxxiii, p. 893.
516 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
the electromotive force of the cell is exactly equal to its diminution
of energy per unit of electricity transmitted. But such a coincidence
is far less significant than the fact that an absorption of heat has been
observed. With acid containing about seven equivalents of water
(HN0 6 -f7HO), M. Favre has found
(V"~ V) Ae — 46781 caL , Z/£z=-41824 caL , A Q — 4957 cah ;
and with acid containing about one equivalent of water (HN0 6 -|-HO),
(F"— F'Me= 49847 ca \ As —— 52714 caL , A $ = -2867 caL .
In the first example, it will be observed that the quantity of heat
absorbed in the cell is not small, and that the electromotive force is
nearly one-eighth greater than can be accounted for by the diminu-
tion of energy in the cell.
This absorption of heat in the cell he has observed in other cases,
in which the chemical processes are much more simple.
For electrodes of cadmium and platinum in hydrochloric acid his
experiments give*
( Y"— V) Ae = 9256 caL , As = — 8258 caL ,
A W v = — 290 caL , A Q = 1288 ca \
In this case the electromotive force is nearly one-sixth greater than
can be accounted for by the diminution of energy in the cell with the
work done against the pressure of the atmosphere.
For electrodes of zinc and platinum in the same acid one series of
experiments givesf
( Y"~ V) Ae = 16950 cal -, As= — 16189 caK ,
Z/IFp = -290 caI -, A Q— 1051 cal -;
and a later series,]]
(F"- V) Ae=16l38™ ] ; As= — 17702 caU ,
AW P = -290™\ J#=-674 ca1 -.
In the electrolysis of hydrochloric acid in a cell with a porous par-
tition, he has found§
* Comptes Rendus, t. Ixviii, p. 1305. The total heat obtained in the whole circuit
(including the cell) when all the electromotive work is turned into heat, was ascer-
tained by direct experiment. This quantity, 7968 calories, is evidently represented by
( Y"- V) Ae- AQ, also by - Ae + A W P . [See (691).] The value of ( V" — V')Ae
is obtained by adding AQ, and that of — Ae by adding — A W P , which is easily esti-
mated, being determined by the evolution of one kilogramme of hydrogen.
\ Ibid.
X Mem. Savants Strang., loc. cit., p. 145.
§ Ibid, p. 142.
J. JV. Gibbs — Equilibrium of Heterogeneous Substances. 51 7
( y _ V") Ae — 34825 caL A Q = 21 13 cal -,
whence
Ae-AW F = 36938.
We cannot assign a precise value to A W F , since the quantity of chlo-
rine which was evolved in the form of gas is not stated. But the
value of —J TF P must lie between 290 CRl - and 580 cah , probably nearer
to the former.
The great difference in the results of the two series of experiments
relating to electrodes of zinc and platinum in hydrochloric acid is
most naturally explained by supposing some difference in the condi-
tions of the experiment, as in the concentration of the acid, or in the
extent to which the substitution of zinc for hydrogen took place.*
That which it is important for us to observe in all these cases is that
there are conditions under which heat is absorbed in a galvanic or
electrolytic cell, so that the galvanic cell has a greater electromotive
force than can be accounted for by the diminution of its energy, and
the operation of electrolysis requires a less electromotive force than
would be calculated from the increase of energy in the cell, — espe-
cially when the work done against the pressure of the atmosphere is
taken into account.
It should be noticed that in all these experiments the quantity rep-
resented by A Q (which is the critical quantity with respect to the
point at issue) was determined by direct measurement of the heat
absorbed or evolved by the cell when placed alone in a calorimeter.
The resistance of the circuit was made so great by a rheostat placed
outside of the calorimeter that the resistance of the cell was regarded
as insignificant in comparison, and no correction appears to have been
made in any case for this resistance. With exception of the error
due to this circumstance, which would in all cases diminish the heat
absorbed in the cell (or increase the heat evolved), the probable error
of A Q must be very small in comparison with that of ( V— V") Ae,
or with that of Ae, which were in general determined by the compar-
* It should perhaps be stated that in his extended memoir published in 1877 in the
Memoires des Savants Grangers, in which he has presumably collected those results
of his experiments which he regards as most important and most accurate, M. Favre
does not mention the absorption of heat in a cell of this kind, or in the similar cell in
which cadmium takes the place of zinc. This may be taken to indicate a decided
preference for the later experiments which showed an evolution of heat. Whatever
the ground of this preference may have been, it can hardly destroy the significance
of the absorption of heat, which was a matter of direct observation in repeated experi-
ments. See Comptes Bendits, t. lxviii, p. 1305.
518 J.W. Gibbs — Equilibrium of Heterogeneous Substances.
ison of different calorimetrical measurements, involving very much
greater quantities of heat.
In considering the numbers which have been cited, we should
remember that when hydrogen is evolved as gas the process is in
general very far from reversible. In a perfect electrochemical appara-
tus, the same changes in the cell would yield a much greater amount
of electromotive work, or absorb a much less amount. In either case,
the value of A Q would be much greater than in the imperfect appara-
tus, the difference being measured perhaps by thousands of calories.*
It often occurs in a galvanic or electrolytic cell that an ion which
is set free at one of the electrodes appears in part as gas, and is in
part absorbed by the electrolytic fluid, and in part absorbed by the
electrode. In such cases, a slight variation in the circumstances,
which would not sensibly affect the electromotive force, would cause
all of the ion to be disposed of in one of the three ways mentioned, if
the current were sufficiently weak. This would make a considerable
* Except in the case of the Grove's cell, in which the reactions are quite complicated,
the absorption of heat is most marked in the electrolysis of hydrochloric acid. The
latter case is interesting, since the experiments confirm the presumption afforded by
the behavior of the substances in other circumstances. (See page 514.) In addition
to the circumstances mentioned above tending to diminish the observed absorption of
heat, the following, which are peculiar to this case, should be noticed.
The electrolysis was performed in a cell with a porous partition, in order to prevent
the chlorine and hydrogen dissolved in the liquid from coming in contact with each
other. It had appeared in a previous series of experiments {Mem. Savants Strang.,
loc. cit., p. 131 ; or Gomptes Bendus, t. lxvi, p. 1231,) that a very considerable amount of
heat might be produced by the chemical union of the gases in solution. In a cell
without partition, instead of an absorption, an evolution of heat took place, which
sometimes exceeded 5000 calories. If, therefore, the partition did not perfectly per-
form its office, this could only cause a diminution in the value of A Q.
A. large part at least of the chlorine appears to have been absorbed by the electro-
lytic fluid. It is probable that a slight difference in the circumstances of the experi-
ment — a diminution of pressure, for example, — might have caused the greater part of
the chlorine to be evolved as gas, without essentially affecting the electromotive force.
The solution of chlorine in water presents some anomalies, and may be attended with
complex reactions, but it appears to be always attended with a very considerable evolu-
tion of heat. (See Berthelot, Oomptes Bendus, t. lxxvi, p. 15 11.) If we regard the evolu-
tion of the chlorine in the form of gas as the normal process, we may suppose that the
absorption of heat in the cell was greatly diminished by the retention of the chlorine
in solution.
Under certain circumstances, oxygen is evolved in the electrolysis of dilute hydro-
chloric acid. It does not appear that this took place to any considerable extent in the
experiments which we are considering. But so far as it may have occurred, we may
regard it as a case of the electrolysis of water. The significance of the fact of the
absorption of heat is not thereby affected.
J. W. G-ibbs — Equilibrium of Heterogeneous Substances. 519
difference in the variation of energy in the cell, and the electromotive
force cannot certainly be calculated from the variation of energy
alone in all these cases. The correction due to the work performed
against the pressure of the atmosphere when the ion is set free as gas
will not help us in reconciling these differences. It will appear on
consideration that this correction will in general increase the discord-
ance in the values of the electromotive force. Nor does it distinctly
appear which of these cases is to be regarded as normal and which
are to be rejected as involving secondary processes.*
If in any case secondary processes are excluded, we should expect
it to be when the ion is identical in substance with the electrode upon
which it is deposited, or from which it passes into the electrolyte.
But even in this case we do not escape the difficulty of the different
forms in which the substance may appear. If the temperature of the
experiment is at the melting point of a metal which forms the ion
and the electrode, a slight variation of temperature will cause the
ion to be deposited in the solid or in the liquid state, or, if the current
is in the opposite direction, to be taken up from a solid or from a
liquid body. Since this will make a considerable difference in the
variation of energy, we obtain different values for the electromotive
force above and beloAV the melting point of the metal, unless we
also take account of the variations of entropy. Experiment does
not indicate the existence of any such difference,! and when we take
account of variations of entropy, as in equation (694), it is apparent
that there ought not to be any, the terms— and t~ ' beino- both
de de &
* It will be observed that in using the formulae (694) and (696) we do not have to
make any distinction between primary and secondary processes. The only limitation
to the generality of these formulae depends upon the reversibility of the processes,
and this limitation does not apply to (696).
f M. Eaoult has experimented with a galvanic element having an electrode of bis-
muth in contact with phosphoric acid containing phosphate of bismuth in solution.
(See Comptes Rendus, t. Ixviii, p. 643.) Since this metal absorbs in melting 12.64
calories per kilogramme or 885 calories per equivalent (70 ki1 -), while a Daniell's cell
yields about 24.000 calories of electromotive work per equivalent of metal, the solid or
liquid state of the bismuth ought to make a difference of electromotive force repre-
sented by .037 of a Daniell's cell, if the electromotive force depended simply upon the
energy of the cell. But in M. Raoult's experiments no sudden change of electromotive
force was manifested at the moment when the bismuth changed its state of aggrega-
tion. In fact, a change of temperature in the electrode from about fifteen degrees
above to about fifteen degrees below the temperature of fusion only occasioned a
variation of electromotive force equal to .002 of a Daniell's cell.
Experiments upon lead and tin gave similar results.
Tbans. Conn. Acad., Vol. III. 66 July, 1878.
520 J. W. Gribbs — Equilibrium of Heterogeneous Substances.
affected by the same difference, viz., the heat of fusion of an electro-
chemical equivalent of the metal. In fact, if such a difference existed,
it would be easy to devise arrangements by which the heat yielded
by a metal in passing from the liquid to the solid state could be
transformed into electromotive work (and therefore into mechanical
work) without other expenditure.
The foregoing examples will be sufficient, it is believed, to show
the necessity of regarding other considerations in determining the
electromotive force of a galvanic or electrolytic cell than the variation
of its energy alone (when its temperature is supposed to remain con-
stant), or corrected only for the work which may be done by external
pressures or by gravity. But the relations expressed by (693), (694),
and (696) may be put in a briefer form.
If we set, as on page 144,
we have, for any constant temperature,
dtp= de — tdr}\
and for any perfect electrochemical apparatus, the temperature of
which is maintained constant,
v „_ vl=z _di dW, dW,
de de de
and for any cell whatever, when the temperature is maintained uni-
form and constant,
(F"~ V')de^ —dtp + dW G +dW F . (698)
In a cell of any ordinary dimensions, the work done by gravity, as
well as the inequalities of pressure in different parts of the cell may
be neglected. If the pressure as well as the temperature is main-
tained uniform and constant, and we set, as on page 147,
C= e - trf + p v,
where p denotes the pressure in the cell, and v its total volume (in-
cluding the products of electrolysis), we have
cfc = ds — tdi]-\- p dv,
and for a perfect electro-chemical apparatus,
V" -V'=-^, (699)
or for any cell,
(r~r)*= - <#• C 700 )
SYNOPSIS OF SUBJECTS TREATED.
Page
Preliminary Remark on the role of energy and entropy in the theory of
thermodynamic systems, 108
CRITERIA OP EQUILIBRIUM AND STABILITY.
Criteria enunciated, _ 109
Meaning of the term possible variations, 110
Passive resistances, . HI
Validity of the criteria, , 112
THE CONDITIONS OF EQUILIBRIUM FOR HETEROGENEOUS MASSES IN CONTACT, WHEN
UNINFLUENCED BY GRAVITY, ELECTRICITY, DISTORTION OF THE SOLID MASSES,
OR CAPILLARY TENSIONS.
Statement of the problem, _ 115
Conditions relating to equilibrium between the initially existing homogeneous
parts of the system, _ ng
Meaning of the term homogeneous, 116
Variation of the energy of a homogeneous mass, H6
Choice of substances to be regarded as components. — Actual and possible
components, ]]g
Deduction of the particular conditions of equilibrium when all parts of the
system have the same components, 118
Definition of the potentials for the component substancss in the various
homogeneous masses, _ 119
Case in which certain substances are only possible components in a part of
the system, 220
Form of the particular conditions of equilibrium when there are relations of
convertibility between the substances which are regarded as the com-
ponents of the different masses, 121
Conditions relating to the possible formation of masses unlike any previously
existing, 124
Very small masses cannot be treated by the same method as those of con-
siderable size, ^129
Sense in which formula (52) may be regarded as expressing the condition
sought, 12 9
Condition (53) is always sufficient for equilibrium, but not always necessary, 131
A mass in which this condition is not satisfied, is at least practically unstable' 133
(This condition is farther discussed under the head of Stability. See p. 156).
Effect of solidity of any part of the system , 134
Effect of additional equations of condition, " 137
Effect of a diaphragm, — equilibrium of osmotic forces, ' '" 138
FUNDAMENTAL EQUATIONS.
Definition and properties, 140
Concerning the quantities ip, %, C, 144
Expression of the criterion of equilibrium by means of the quantity"^,"----- 145
Expression of the criterion of equilibrium in certain cases by means of the
quantity J, U>J
POTENTIALS.
The value of a potential for a substance in a given mass is not dependent on the
other substances which may be chosen to represent the composition of the
mass, .j^g
Potentials defined so as to render this property evident, ....... " 149
522 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
Page
In the same homogeneous mass we may distinguish the potentials for an indefinite
number of substances, each of which has a perfectly determined value. Between
the potentials for different substances in the same homogeneous mass the same
equations will subsist as between the units of these substances, 149
The values of potentials depend upon the arbitrary constants involved in the defi-
nition of the energy and entropy of each elementary substance, 151
COEXISTENT PHASES.
Definition of phases — of coexistent phases, _ 152
Number of the independent variations which are possible in a system of coexistent
phases, 152
Case of n+ 1 coexistent phases,-. 153
Cases of a less number of coexistent phases, 155
INTERNAL STABILITY OP HOMOGENEOUS FLUIDS AS INDICATED BY FUNDAMENTAL
EQUATIONS.
General condition of absolute stability, 156
Other forms of the condition, . 160
Stability in respect to continuous changes of phase, 162
Conditions which characterize the limits of stability in this respect, 169
GEOMETRICAL ILLUSTRATIONS.
Surfaces in which the composition of the body represented is constant, 172
Surfaces and curves in which the composition of the body represented is variable
and its temperature and pressure are constant, 176
CRITICAL PHASES.
Definition, 188
Number of independent variations which are possible for a critical phase while
remaining such, 188
Analytical expression of the conditions which characterize critical phases. — Situ-
ation of critical phases with respect to the limits of stability, '. 189
Variations which are possible under different circumstances in the condition of a
mass initially in a critical phase, _ - _ --- 191
ON THE VALUES OF THE POTENTIALS WHEN THE QUANTITY OF ONE OF THE
■COMPONENTS IS VERY SMALL, 194
ON CERTAIN POINTS RELATING TO THE MOLECULAR CONSTITUTION OF BODIES.
Proximate and ultimate components, - 197
Phases of dissipated energy, 200
Catalysis. — perfect catalytic agent, 201
A. fundamental equation for phases of dissipated energy may be formed from the
more general form of the fundamental equation, 201
The phases of dissipated energy may sometimes be the only phases the existence
of which can be experimentally verified, 201
THE CONDITIONS OF EQUILIBRIUM FOR HETEROGENEOUS MASSES UNDER THE INFLU-
ENCE OF GRAVITY.
The problem is treated by two different methods :
The elements of volume are regarded as variable, 203
The elements of volume are regarded as fixed, . 207
FUNDAMENTAL EQUATIONS OF IDEAL GASES AND GAS-MIXTURES.
Ideal gas, 210
Ideal gas-mixture — Dalton's Law, -- 215
Inferences in regard to potentials in liquids and solids, . _ . 225
Considerations relating to the increase of entropy due to the mixture of gases by
diffusion, --.-- ' 2 '- jt
The phases of dissipated energy of an ideal gas-mixture with components which
are chemically related, 230
J. W. Gibbs — Equilibrium of Heterogeneous Substances. 523
Page
Gas-mixtures with convertible components, 234
Case of peroxide of nitrogen, 237
Fundamental equations for the phases of equilibrium, 245
SOLIDS.
The conditions of internal and external equilibrium for solids in contact with fluids
with regard to all possible states of strain, 343
Strains expressed by nine differential coefficients, 344
Variation of energy in an element of a solid, 344
Deduction of the conditions of equilibrium, 346
Discussion of the condition which relates to the dissolving of the solid, 352
Fundamental equations for solids. 361
Concerning solids which absorb fluids, 375
THEORY OP CAPILLARITY.
SURFACES OF DISCONTINUITY BETWEEN FLUID MASSES.
Preliminary notions. — Surfaces of discontinuity. — Dividing surface, 380
Discussion of the problem. — The particular conditions of equilibrium for contigu-
ous masses relating to temperature and the potentials which have already been
obtained are not invalidated by the influence of the surface of discontinuity. —
Superficial energy and entropy. — Superficial densities of the component sub-
stances. — General expression for the variation of the superficial energy. — Con-
dition of equilibrium relating to the pressures in the contiguous masses, 380
Fundamental equations for surfaces of discontinuity between fluid masses, 391
Experimental determination of the same, 394
Fundamental equations for plane surfaces, 395
Stability of surfaces of discontinuity —
(1 ) with respect to changes in the nature of the surface, 400
(2) with respect to changes in which the form of the surface is varied, 405
On the possibility of the formation of a fluid of different phase within any homo-
geneous fluid, _ 426
On the possible formation at the surface where two different homogeneous fluids
meet of a fluid of different phase from either, _ 422
Substitution of pressures for potentials in fundamental equations for surfaces 429
Thermal and mechanical relations pertaining to the extension of surfaces of dis-
continuity, 434
Impermeable films, , _ " 44Q
The conditions of internal equilibrium for a system of heterogeneous fluid masses
without neglect of the influence of the surfaces of discontinuity or of gravity . 442
Conditions of stability, ' 45^
On the possibility of the formation of a new surface of discontinuity where sev-
eral surfaces of discontinuity meet, _ 453
The conditions of stability for fluids relating to the formation of a new phase at" a
line in which three surfaces of discontinuity meet, ' 455
The conditions of stability for fluids relating to the formation of a new phase at" a
point where the vertices of four different masses meet, 46 4
Liquid films, " j™
Definition of an element of the film, " ^gn
Each element may generally be regarded as in a state of equilibrium. Prop-
erties of an element in such a state and sufficiently thick for its interior to
have the properties of matter in mass. — Conditions under which an exten-
sion of the film will not cause an increase of tension.— When the film has
more than one component which does not belong to the contiguous masses
extension will in^general cause an increase of tension. — Value of the elas-
ticity of the film' deduced from the fundamental equations of the surfaces
and masses. — Elasticity manifest to observation, 46g
The elasticity of a film does not vanish at the limit at which its "interior
ceases to have the properties of matter in mass, but a certain kind of
instability is developed, 4*0
Application of the conditions of equilibrium already deduced for a" system
under the influence of gravity (pages 447, 448) to the case of a liquid film 473
Concerning the formation of liquid films and the processes which lead to
their destruction.— Black spots in films of soap-water, 475
524 J. W. Gibbs — Equilibrium of Heterogeneous Substances.
SURFACES OF DISCONTINUITY BETWEEN SOLIDS AND FLUIDS.
Page
Preliminary notions, 482
Conditions of equilibrium for isotropic solids, 485
Effect of gravity, 488
Conditions of equilibrium in the case of crystals, 489
Effect of gravity, 492
Limitations, 493
Conditions of equilibrium for a line at which three different masses meet, one of
which is solid, 495
General relations, - 497
Another method and notation, 497
ELECTROMOTIVE FORCE.
Modification of the conditions of equilibrium by electromotive force, 501
Equation of fluxes. — Ions. — Electro-chemical equivalents, 501
Conditions of equilibrium, 502
Pour cases, . 503
Lippmann's electrometer, 506
Limitations due to passive resistances, 507
General properties of a perfect electro-chemical apparatus, 508
Eeversibility the test of perfection, .. 508
Determination of the electromotive force from the changes which take place
in the cell. — Modification of the formula for the case of an imperfect
apparatus, 509
When the temperature of the cell is regarded as constant, it is not allowable
to neglect the variation of entropy due to heat absorbed or evolved. — This
is shown by a Grove's gas battery charged with hydrogen and nitrogen, 510
by the currents caused by differences in the concentration of the electrolyte, 511
and by electrodes of zinc and mercury in a solution of sulphate of zino,__, 511
That the same is true when the chemical processes take place by definite
proportions is shown by a priori considerations based on the phenomena
exhibited in the direct combination of the elements of water or of hydro-
chloric acid, - 513
and by the absorption of heat which M. Pavre has in many cases observed
in a galvanic or electrolytic cell, 516
The different physical states in which the ion is deposited do not affect the
value of the electromotive force, if the phases are coexistent. — Experiments
of M. Raoult, 518
Other formulae for the electromotive force, 520
ERRATA.
Page 356, last line but two, for crystalline solid, read solid of continuous crystalline
structure.
Page 385, line 13, for M', read M.
Pages 391, 394, 395, 400, in headings, after Discontinuity, add between Fluid Masses.
Page 403, line 16, after any other film, add of the same components.
Page 405, line 29, after this, add case.
Page 432, line 15 of foot-note, for H, read H,.
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